Question 1: Palette 1/197

Allen JEE (Advanced) 2023 MCQ

A bar of mas $M = 1.00 \, kg$ and length $L = 0.20 \, m$ is lying on a horizontal frictionless surface. One end of the bar is pivoted at a point about which it is free to rotate. A small mass $m = 0.10 \, kg$ is moving on the same horizontal surface with $5.00 \, ms^{-1}$ speed on a path perpendicular to the bar. It hits the bar at a distance $L/2$ from the pivoted end and returns back on the same path with speed $v$ . After this elastic collision, the bar rotates with an angular velocity $\omega$ . Which of the following statement is correct?

A

$\omega = 6.98\mathrm{rad}s^{-1}$ and $v = 4.30ms^{-1}$

B

$\omega = 3.75$ rad $s^{-1}$ and $v = 4.30$ ms $^{-1}$

C

$\omega = 3.75$ rad $s^{-1}$ and $v = 10.0$ ms $^{-1}$

D

$\omega = 6.80$ rad $s^{-1}$ and $v = 4.10$ ms $^{-1}$

Question 2: Palette 2/197

Allen JEE (Advanced) 2023 MSQ

An annular disk of mass M, inner radius a and outer radius b is placed on a horizontal surface with coefficient of friction $\mu$ , as shown in the figure. At some time, an impulse $J_{0}\hat{x}$ is applied at a height h above the center of the disk. If $h = h_{m}$ then the disk rolls without slipping along the x-axis. Which of the following statement(s) is(are) correct?

A

For $\mu \neq 0$ and $a \to 0$ , $h_m = b / 2$

B

For $\mu \neq 0$ and $a \to b$ , $h_m = b$

C

For $h = h_{m}$ , the initial angular velocity does not depend on the inner radius a.

D

For $\mu = 0$ and h = 0, the wheel always slides without rolling.

Question 3: Palette 3/197

Allen JEE (Advanced) 2022 MCQ

A flat surface of a thin uniform disk A of radius R is glued to a horizontal table. Another thin uniform disk B of mass M and with the same radius R rolls without slipping on the circumference of A, as shown in the figure. A flat surface of B also lies on the plane of the table. The center of mass of B has fixed angular speed $\omega$ about the vertical axis passing through the center of A. The angular momentum of B is $nM\omega R^{2}$ with respect to the center of A. Which of the following is the value of n?

A

2

B

5

C

$\frac{7}{2}$

D

$\frac{9}{2}$

Question 4: Palette 4/197

Allen JEE (Main) 2021 MCQ

A triangular plate is shown. A force $\vec{F} = 4\hat{i} - 3\hat{j}$ is applied at point P. The torque at point P with respect to point 'O' and 'Q' are :

A

$- 15 - 20 \sqrt{3}$ , $15 - 20 \sqrt{3}$

B

$15 + 20\sqrt{3}, 15 - 20\sqrt{3}$

C

$15 - 20\sqrt{3}, 15 + 20\sqrt{3}$

D

$-15 + 20\sqrt{3}, 15 + 20\sqrt{3}$

Question 5: Palette 5/197

Allen JEE (Main) 2021 MCQ

A solid cylinder of mass m is wrapped with an inextensible light string and, is placed on a rough inclined plane as shown in the figure. The frictional force acting between the cylinder and the inclined plane is: [The coefficient of static friction, $\mu_{s}$ , is 0.4]

A

$\frac{7}{2}mg$

B

5 mg

C

$\frac{mg}{5}$

D

0

Question 6: Palette 6/197

Allen JEE (Main) 2021 MCQ

Two discs have moments of inertia $I_{1}$ and $I_{2}$ about their respective axes perpendicular to the plane and passing through the centre. They are rotating with angular speeds, $\omega_{1}$ and $\omega_{2}$ respectively and are brought into contact face to face with their axes of rotation coaxial. The loss in kinetic energy of the system in the process is given by:

A

$\frac{I_{1}I_{2}}{(I_{1}+I_{2})}\left(\omega_{1}-\omega_{2}\right)^{2}$

B

$\frac{\left(I_{1}-I_{2}\right)^{2}\omega_{1}\omega_{2}}{2\left(I_{1}+I_{2}\right)}$

C

$\frac{I_{1}I_{2}}{2\left(I_{1}+I_{2}\right)}\left(\omega_{1}-\omega_{2}\right)^{2}$

D

$\frac{\left(\omega_{1}-\omega_{2}\right)^{2}}{2\left(I_{1}+I_{2}\right)}$

Question 7: Palette 7/197

Allen JEE (Main) 2021 MCQ

The solid cylinder of length 80 cm and mass M has a radius of 20 cm. Calculate the density of the material used if the moment of inertia of the cylinder about an axis CD parallel to AB as shown in figure is 2.7 kg m $^{2}$ .

A

$14.9 \, kg / m^{3}$

B

$7.5 \times 10^{1} \, kg / m^{3}$

C

$7.5 \times 10^{2} \, kg/m^{3}$

D

$1.49 \times 10^{2} \, kg / m^{3}$

Question 8: Palette 8/197

Allen JEE (Advanced) 2021 MSQ

A horizontal force F is applied at the center of mass of a cylindrical object of mass m and radius R, perpendicular to its axis as shown in the figure. The coefficient of friction between the object and the ground is $\mu$ . The center of mass of the object has an acceleration a. The acceleration due to gravity is g. Given that the object rolls without slipping, which of the following statement(s) is(are) correct?

A

For the same F, the value of a does not depend on whether the cylinder is solid or hollow.

B

For a solid cylinder, the maximum possible value of $a$ is $2\mu g$ .

C

The magnitude of the frictional force on the object due to the ground is always $\mu mg$ .

D

For a thin-walled hollow cylinder, $a = \frac{F}{2m}$ .

Question 9: Palette 9/197

Allen JEE (Advanced) 2021 MSQ

A particle of mass M = 0.2 kg is initially at rest in the xy-plane at a point $(x = -l, y = -h)$ , where l = 10 m and h = 1m. The particle is accelerated at time t = 0 with a constant acceleration $a = 10 \, m/s^{2}$ along the positive x-direction. Its angular momentum and torque with respect to the origin, in SI units, are represented by $\vec{L}$ and $\vec{\tau}$ , respectively. $\hat{i}, \hat{j}$ and $\hat{k}$ are unit vectors along the positive x, y and z-directions, respectively. If $\hat{k} = \hat{i} \times \hat{j}$ then which of the following statement(s) is(are) correct?

A

The particle arrives at the point $(x = l, y = -h)$ at time t = 2s

B

$\vec{\tau} = 2\hat{k}$ when the particle passes through the point (x = l, y = -h)

C

$\vec{L} = 4\hat{k}$ when the particle passes through the point $(x = l, y = -h)$

D

$\vec{\tau} = \hat{k}$ when the particle passes through the point $(x = 0, y = -h)$

Question 10: Palette 10/197

Allen JEE (Advanced) 2021 MSQ

One end of a horizontal uniform beam of weight W and length L is hinged on a vertical wall at point O and its other end is supported by a light inextensible rope. The other end of the rope is fixed at point Q, at a height L above the hinge at point O. A block of weight $\alpha W$ is attached at the point P of the beam, as shown in the figure (not to scale). The rope can sustain a maximum tension of $(2\sqrt{2})W$ . Which of the following statement(s) is(are) correct?

A

The vertical component of reaction force at O does not depend on $\alpha$ .

B

The horizontal component of reaction force at O is equal to W for $\alpha = 0.5$ .

C

The tension in the rope is 2W for $\alpha = 0.5$ .

D

The rope breaks if $\alpha > 1.5$ .

Question 11: Palette 11/197

Allen JEE (Main) 2020 MCQ

A uniform cylinder of mass M and radius R is to be pulled over a step of height a (a < R) by applying a force F at its centre 'O' perpendicular to the plane through the axes of the cylinder on the edge of the step (see figure). The minimum value of F required is:

A

$Mg\sqrt{1-\frac{a^{2}}{R^{2}}}$

B

$Mg\sqrt{\left(\frac{R}{R-a}\right)^{2}-1}$

C

$Mg\frac{a}{R}$

D

$Mg\sqrt{1-\left(\frac{R-a}{R}\right)^{2}}$

Question 12: Palette 12/197

Allen JEE (Main) 2020 MCQ

A block of mass m = 1 kg slides with velocity v = 6 m/s on a frictionless horizontal surface and collides with a uniform vertical rod and sticks to it as shown. The rod is pivoted about O and swings as a result of the collision making angle $\theta$ before momentarily coming to rest. If the rod has mass M = 2 kg, and length l = 1 m, the value of $\theta$ is approximately : (Take g = 10 m/s $^{4}$ )

A

$6 9 ^ {\circ}$

B

$6 3 ^ {\circ}$

C

$5 5 ^ {\circ}$

D

$4 9 ^ {\circ}$

Question 13: Palette 13/197

Allen JEE (Main) 2020 MCQ

Mass per unit area of a circular disc of radius a depends on the distance r from its centre as $\sigma(r) = A + Br$ . The moment of inertia of the disc about the axis, perpendicular to the plane and passing through its centre is :

A

$2\pi a^{4}\left(\frac{A}{4}+\frac{aB}{5}\right)$

B

$\pi a^{4}\left(\frac{A}{4}+\frac{aB}{5}\right)$

C

$2\pi a^{4}\left(\frac{aA}{4}+\frac{B}{5}\right)$

D

$2\pi a^{4}\left(\frac{A}{4}+\frac{B}{5}\right)$

Question 14: Palette 14/197

Allen JEE (Advanced) 2020 MCQ

A football of radius R is kept on a hole of radius r (r < R) made on a plank kept horizontally. One end of the plank is now lifted so that it gets tilted making an angle $\theta$ from the horizontal as shown in the figure below. The maximum value of $\theta$ so that the football does not start rolling down the plank satisfies (figure is schematic and not drawn to scale) –

A

$\sin \theta = \frac{r}{R}$

B

$\tan \theta = \frac{r}{R}$

C

$\sin \theta = \frac{r}{2R}$

D

$\cos\theta=\frac{r}{2R}$

Question 15: Palette 15/197

Allen JEE (Advanced) 2020 MCQ

A small roller of diameter 20 cm has an axle of diameter 10 cm (see figure below on the left). It is on a horizontal floor and a meter scale is positioned horizontally on its axle with one edge of the scale on top of the axle (see figure on the right). The scale is now pushed slowly on the axle so that it moves without slipping on the axle, and the roller starts rolling without slipping. After the roller has moved 50 cm, the position of the scale will look like (figures are schematic and not drawn to scale)-

A

C

Question 16: Palette 16/197

Allen JEE (Main) 2019 MCQ

To mop-clean a floor, a cleaning machine presses a circular mop of radius R vertically down with a total force F and rotates it with a constant angular speed about its axis. If the force F is distributed uniformly over the mop and if coefficient of friction between the mop and the floor is $\mu$ , the torque, applied by the machine on the mop is :

A

$\frac{2}{3}\mu FR$

B

$\mu FR/3$

C

$\mu FR/2$

D

$\mu FR/6$

Question 17: Palette 17/197

Allen JEE (Main) 2019 MCQ

An equilateral triangle ABC is cut from a thin solid sheet of wood. (see figure) D, E and F are the mid-points of its sides as shown and G is the centre of the triangle. The moment of inertia of the triangle about an axis passing through G and perpendicular to the plane of the triangle is $I_{0}$ . If the smaller triangle DEF is removed from ABC, the moment of inertia of the remaining figure about the same axis is I. Then:

A

$I = \frac{9}{16} I_0$

B

$I = \frac{3}{4} I_0$

C

$I = \frac{I_0}{4}$

D

$I = \frac{15}{16} I_0$

Question 18: Palette 18/197

Allen JEE (Main) 2019 MCQ

A rectangular solid box of length 0.3 m is held horizontally, with one of its sides on the edge of a platform of height 5m. When released, it slips off the table in a very short time $\tau = 0.01s$ , remaining essentially horizontal. The angle by which it would rotate when it hits the ground will be (in radians) close to :-

A

0.02

B

0.28

C

0.5

D

0.3

Question 19: Palette 19/197

Allen JEE (Main) 2019 MCQ

A stationary horizontal disc is free to rotate about its axis. When a torque is applied on it, its kinetic energy as a function of $\theta$ , where $\theta$ is the angle by which it has rotated, is given as $k\theta^2$ . If its moment of inertia is $I$ then the angular acceleration of the disc is:

A

$\frac{k}{2I}\theta$

B

$\frac{k}{I}\theta$

C

$\frac{k}{4I}\theta$

D

$\frac{2k}{I}\theta$

Question 20: Palette 20/197

Allen JEE (Advanced) 2018 MSQ

The potential energy of a particle of mass m at a distance r from a fixed point O is given by $V(r) = kr^{2}/2$ , where k is a positive constant of appropriate dimensions. This particle is moving in a circular orbit of radius R about the point O. If v is the speed of the particle and L is the magnitude of its angular momentum about O, which of the following statements is (are) true?

A

$v = \sqrt{\frac{k}{2m}} R$

B

$v = \sqrt{\frac{k}{m}} R$

C

$L = \sqrt{mk} R^{2}$

D

$L = \sqrt{\frac{mk}{2}} R^{2}$

Question 21: Palette 21/197

Allen JEE (Main) 2017 MCQ

The moment of inertia of a uniform cylinder of length $\ell$ and radius $R$ about its perpendicular bisector is $I$ . What is the ratio $\ell / R$ such that the moment of inertia is minimum?

A

1

B

$\frac{3}{\sqrt{2}}$

C

$\sqrt{\frac{3}{2}}$

D

$\frac{\sqrt{3}}{2}$

Question 22: Palette 22/197

Allen JEE (Advanced) 2017 MSQ

A rigid uniform bar AB of length L is slipping from its vertical position on a frictionless floor (as shown in the figure). At some instant of time, the angle made by the bar with the vertical is $\theta$ . Which of the following statements about its motion is/are correct?

A

When the bar makes an angle $\theta$ with the vertical, the displacement of its midpoint from the initial position is proportional to $(1 - \cos \theta)$ .

B

The midpoint of the bar will fall vertically downward.

C

Instantaneous torque about the point in contact with the floor is proportional to $\sin\theta$ .

D

The trajectory of the point A is a parabola.

Question 23: Palette 23/197

Allen JEE (Advanced) 2017 MCQ

A wheel of radius R and mass M is placed at the bottom of a fixed step of height R as shown in the figure. A constant force is continuously applied on the surface of the wheel so that it just climbs the step without slipping. Consider the torque $\tau$ about an axis normal to the plane of the paper passing through the point Q. Which of the following options is/are correct?

A

If the force is applied normal to the circumference at point X then $\tau$ is constant.

B

If the force is applied tangentially at point S then $\tau \neq 0$ but the wheel never climbs the step.

C

If the force is applied normal to the circumference at point P then $\tau$ is zero.

D

If the force is applied at point P tangentially then $\tau$ decreases continuously as the wheel climbs.

Question 24: Palette 24/197

Allen JEE (Main) 2016 MCQ

A particle of mass m is moving along the side of a square of side 'a', with a uniform speed $\upsilon$ in the x-y plane as shown in the figure. Which of the following statement is false for the angular momentum $\vec{L}$ about the origin?

A

$\vec{L} = \frac{mv}{\sqrt{2}} R\hat{k}$ when the particle is moving from $D$ to $A$

B

$\vec{L} = -\frac{mv}{\sqrt{2}} R\hat{k}$ when the particle is moving from $A$ to $B$

C

$\vec{L} = mv\left[\frac{R}{\sqrt{2}} - a\right]\hat{k}$ when the particle is moving from $C$ to $D$

D

$\vec{L} = mv\left[\frac{R}{\sqrt{2}} + a\right]\hat{k}$ when the particle is moving from $B$ to $C$

Question 25: Palette 25/197

Allen JEE (Advanced) 2016 MSQ

The position vector of a particle of mass m is given by the following equation $\vec{r}(t) = \alpha t^{3}\hat{i} + \beta t^{2}\hat{j}$ , where $\alpha = \frac{10}{3}ms^{-3}$ , $\beta = 5ms^{-2}$ and m = 0.1 kg. At t = 1 s, which of the following statement(s) is(are) true about the particle?

A

The velocity $\vec{v}$ is given by $\vec{v} = (10\hat{i} + 10\hat{j})ms^{-1}$ .

B

The angular momentum $\vec{L}$ with respect to the origin is given by $\vec{L} = -\left(\frac{5}{3}\right)\hat{k}$ . Nms.

C

The force $\vec{F}$ is given by $\vec{F} = (\hat{i} + 2\hat{j})N$ .

D

The torque $\vec{\tau}$ with respect to the origin is given by $\vec{\tau} = -\left(\frac{20}{3}\right)\hat{k}$ $Nm$ .

Question 26: Palette 26/197

Allen JEE (Advanced) 2016 MCQ

Comprehension Passage: PARAGRAPH FOR QUESTION NO. 5 AND 6

A frame of reference that is accelerated with respect to an inertial frame of reference is called a non-inertial frame of reference. A coordinate system fixed on a circular disc rotating about a fixed axis with a constant angular velocity $\omega$ is an example of a non-inertial frame of reference. The relationship between the force $\vec{F}_{rot}$ experienced by a particle of mass m moving on the rotating disc and the force $\vec{F}_{in}$ experienced by the particle in an inertial frame of reference is $\vec{F}_{rot} = \vec{F}_{in} + 2m(\vec{v}_{rot} \times \vec{\omega}) + m(\vec{\omega} \times \vec{r}) \times \vec{\omega}$ , where $\vec{v}_{rot}$ is the velocity of the particle in the rotating frame of reference and $\vec{r}$ is the position vector of the particle with respect to the centre of the disc. Now consider a smooth slot along a diameter of a disc of radius R rotating counter-clockwise with a constant angular speed $\omega$ about its vertical axis through its center. assign a coordinate system with the origin at the centre of the disc, the x-axis along the slot, the y-axis perpendicular to the slot and the z-axis along the rotation axis ( $\vec{\omega} = \omega\hat{k}$ ). A small block of mass m is gently placed in the slot at $\vec{r} = (R/2)\hat{i}$ at t = 0 and is constrained to move only along the slot. The distance r of the block at time t is:

A

$\frac{R}{4}(e^{2\omega t} + e^{-2\omega t})$

B

$\frac{R}{2}\cos2\omega t$

C

$\frac{R}{2}\cos\omega t$

D

$\frac{R}{4}(e^{\omega t} + e^{-\omega t})$

Question 27: Palette 27/197

Allen JEE (Advanced) 2016 MCQ

Comprehension Passage: PARAGRAPH FOR QUESTION NO. 5 AND 6

A frame of reference that is accelerated with respect to an inertial frame of reference is called a non-inertial frame of reference. A coordinate system fixed on a circular disc rotating about a fixed axis with a constant angular velocity $\omega$ is an example of a non-inertial frame of reference. The relationship between the force $\vec{F}_{rot}$ experienced by a particle of mass m moving on the rotating disc and the force $\vec{F}_{in}$ experienced by the particle in an inertial frame of reference is $\vec{F}_{rot} = \vec{F}_{in} + 2m(\vec{v}_{rot} \times \vec{\omega}) + m(\vec{\omega} \times \vec{r}) \times \vec{\omega}$ , where $\vec{v}_{rot}$ is the velocity of the particle in the rotating frame of reference and $\vec{r}$ is the position vector of the particle with respect to the centre of the disc. Now consider a smooth slot along a diameter of a disc of radius R rotating counter-clockwise with a constant angular speed $\omega$ about its vertical axis through its center. assign a coordinate system with the origin at the centre of the disc, the x-axis along the slot, the y-axis perpendicular to the slot and the z-axis along the rotation axis ( $\vec{\omega} = \omega\hat{k}$ ). A small block of mass m is gently placed in the slot at $\vec{r} = (R/2)\hat{i}$ at t = 0 and is constrained to move only along the slot. The net reaction of the disc on the block is:

A

$-m\omega^{2}R \cos \omega t\hat{j} - mg\hat{k}$

B

$m\omega^{2}R \sin \omega t\hat{j} - mg\hat{k}$

C

$\frac{1}{2}m\omega^{2}R\left(e^{\omega t} - e^{-\omega t}\right)\hat{j} + mg\hat{k}$

D

$\frac{1}{2}m\omega^{2}R\left(e^{2\omega t} - e^{-2\omega t}\right)\hat{j} + mg\hat{k}$

Question 28: Palette 28/197

Allen JEE (Main) 2014 MCQ

A mass ‘m’ is supported by a massless string wound around a uniform hollow cylinder of mass m and radius R. If the string does not slip on the cylinder, with what acceleration will the mass fall on release?

A

$\frac{5g}{6}$

B

$g$

C

$\frac{2g}{3}$

D

$\frac{g}{2}$

Question 29: Palette 29/197

Allen MCQ

A pulley is hinged at the centre and a massless thread is wrapped around it. The thread is pulled with a constant force F starting from rest. As time increases,

A

its angular velocity increases, but force on hinge remains constant

B

its angular velocity remains same, but force on hinge increases

C

its angular velocity increases and force on hinge increases

D

its angular velocity remains same and force on hinge is constant

Question 30: Palette 30/197

Allen MCQ

In the previous question, if $dv/dt = 0$ , then the angular acceleration of the ladder when $\alpha = 45^\circ$ is:

A

$2v^2/L^2$

B

$v^2/2L^2$

C

$\sqrt{2}\left[\frac{v^2}{L^2}\right]$

D

None

Question 31: Palette 31/197

Allen MCQ

A smooth pulley $P$ of mass $M = 2kg$ is lying on a smooth table. A light string passes round the pulley and has mass $m = 1kg$ and $m' = 2kg$ attached to its ends, the two portions of the string being perpendicular to the edge of the table so that the masses hang vertically. Acceleration of pulley is-

A

$\frac{g}{2}$

B

$\frac{2g}{3}$

C

$\frac{4g}{9}$

D

$\frac{4g}{7}$

Question 32: Palette 32/197

Allen MCQ

A fixed pulley of radius 20 cm and moment of inertia $0.32 \, kg.m^{2}$ about its axle has a massless cord wrapped around its rim. A mass M of 2 kg is attached to the end of the cord. The pulley can rotate about its axis without any friction. The acceleration of the mass M is :- (Assume $g = 10 \, m/s^{2}$ )

A

$1m / s^{2}$

B

$3m / s^{2}$

C

$2m / s^{2}$

D

$4m / s^{2}$

Question 33: Palette 33/197

Allen MCQ

A cube with sides of length 2a and mass M is moving with an initial speed $v_{0}$ along a frictionless table. When the cube reaches the end of the table it is caught abruptly by a short lip and begins to rotate. The minimum speed $v_{0}$ such that the cube falls off the table is :-

A

$\sqrt{16ag(\sqrt{2}-1)/3}$`

B

$\sqrt{16ag(\sqrt{2} - 1)}$

C

$\sqrt{ag\left(\sqrt{2} - 1\right)}$

D

$\sqrt{\frac{ag}{3} (\sqrt{2} - 1)}$

Question 34: Palette 34/197

Allen MCQ

As shown in the figure, a uniform rod can be rotated freely about point A in the vertical plane. When a vertical upward force F lifts up the rod slowly from position shown, which of the following statements is correct till the rod becomes horizontal?

A

The pulling force is unchanged, its moment of force increases.

B

The pulling force is unchanged, its moment of force decreases.

C

The pulling force increases, its moment of force decreases.

D

The pulling force decreases, its moment of force decreases.

Question 35: Palette 35/197

Allen MCQ

In the given figure a uniform wheel of radius 30cm rests against a rigid rectangular block 15cm high. All the surfaces are smooth. The wheel weighs 1000 N. The minimum pull P through the center which will turn the wheel over the block is :-

A

$500\sqrt{3} N$

B

$1000\sqrt{3} N$

C

$1000N$

D

$400\sqrt{3} N$

Question 36: Palette 36/197

Allen MCQ

A wheel (disc of mass m, radius R) is mounted on a base of mass M. The whole system is placed on rough horizontal surface. The coefficient of friction between the base & the surface is $\mu$ . Initially the system is at rest. A time varying horizontal force F = 2t is applied tangentially on the wheel, where t is time in second & force F is in newton. The kinetic energy of the disc when the system is about to slide on ground is :-

A

$\frac{1}{m}\left(\frac{\mu(M)g}{2}\right)^4$

B

$\frac{1}{m}\left(\frac{\mu(m+M)g}{2}\right)^{4}$

C

$\frac{1}{m}\left(\frac{\mu(m+M)g}{2}\right)^{2}$

D

$\frac{1}{m}\left(\frac{\mu(M)g}{2}\right)^2$

Question 37: Palette 37/197

Allen MCQ

A uniform disc of mass M and radius R is smoothly pivoted at 'O'. A light inextensible strings wrapped over the disc hangs a particle of mass m. If the system is released from rest, assuming string does not slide, what will be the angular speed of the disc as a function of time.

A

$\omega = \frac{2mgt}{(M + 2m)R}$

B

$\omega = \frac{4mgt}{(M + 2m)R}$

C

$\omega = \frac{mgt}{(M + 2m)R}$

D

$\omega = \frac{mgt}{(M + m)R}$

Question 38: Palette 38/197

Allen MCQ

On a disc of radius R a concentric circle of radius R/2 is drawn. The disc is free to rotate about a frictionless axis through its centre and perpendicular to plane of the disc. All the three forces (in plane of the disc) shown in figure are act tangentially to their respective circular periphering. The magnitude of the net torque (about centre of disc) acting on the disc is :-

A

1.9 FR

B

2.3 FR

C

2.5 FR

D

1.5 FR

Question 39: Palette 39/197

Allen MCQ

Eight point objects, each of mass M, are at the vertices of a rigid, massless framework in the form of a cube; each side is of length a. What is the moment of inertia of this arrangement about an axis passing through one edge of the cube?

A

$4Ma^{2}$

B

$6Ma^{2}$

C

$8Ma^{2}$

D

$10Ma^{2}$

Question 40: Palette 40/197

Allen MCQ

A point mass $m_{A}$ is connected to a point mass $m_{B}$ by a massless rod of length $\ell$ as shown in the figure. It is observed that the ratio of the moment of inertia of the system about the two axes BB and AA, which is parallel to each other and perpendicular to the rod is $\frac{I_{BB}}{I_{AA}} = 3$ . The distance of the centre of mass of the system from the point mass $m_{A}$ is

A

$(3 / 4)\ell$

B

$(2 / 3)\ell$

C

$(1 / 2)\ell$

D

$(1 / 4)\ell$

Question 41: Palette 41/197

Allen MCQ

The square structure shown in figure consists of four point masses connected by rods of negligible mass. Mark the CORRECT option regarding the moment of inertia of the structure about the following axes:

A

Axis A, passing through the centre of the structure and normal to its plane $I_{A} = ma^{2}$

B

Axis B, passing through one of the point masses and normal to the plane of the structure $I_{B} = 3ma^{2}$

C

Axis $CC'$ , passing through two adjacent point masses $I_{\mathrm{C}} = 2ma^{2}$ and

D

Axis $DD'$ , along the diagonal of the structure $I_{\mathrm{D}} = 2ma^{2}$

Question 42: Palette 42/197

Allen MCQ

Nine point mass m are arranged in an double-cone system as shown in diagram. Radius of each cone is R and height of each cone is H. The moment of inertia about the x-axis passing through the common vertex is :

A

$8mH^{2}$

B

$8m(H^{2} + R^{2})$

C

$8mH^{2} + 4mR^{2}$

D

None of these

Question 43: Palette 43/197

Allen MCQ

A rigid body can be hinged about any point on the x-axis. When it is hinged such that the hinge is at x, the moment of inertia is given by $I = 2x^{2} - 12x + 27$ . The x-coordinate of centre of mass is

A

x = 2

B

x = 0

C

x = 1

D

x = 3

Question 44: Palette 44/197

Allen MCQ

The figure shows a uniform rod lying along the x-axis. The locus of all the points lying on the xy-plane, about which the moment of inertia of the rod is same as that about O is :-

A

an ellipse

B

a circle

C

a parabola

D

a straight line

Question 45: Palette 45/197

Allen MCQ

A thin uniform rod of mass $M$ and length $L$ has its moment of inertia $I_{1}$ about its perpendicular bisector. The rod is bend in the form of a semi-circular arc. Now its moment of inertia through the centre of the semi-circular arc and perpendicular to its plane is $I_{2}$ . The ratio of $I_{1}: I_{2}$ will be ____.

A

< 1

B

> 1

C

= 1

D

can't be said

Question 46: Palette 46/197

Allen MCQ

A circular disc X of radius R is made from an iron plate of thickness t and another disc Y of radius 4R is made from an iron plate of thickness t/4. Then the relation between the moment of inertia $I_{X}$ and $I_{Y}$ is

A

$I_{Y} = 32 I_{X}$

B

$I_{Y} = 16 I_{X}$

C

$I_{Y} = I_{X}$

D

$I_{Y} = 64 I_{X}$

Question 47: Palette 47/197

Allen MCQ

A thin wire of length L and uniform linear mass density ρ is bent into a circular loop with centre at O as shown. The moment of inertia of the loop about the axis XX' is

A

ρL³/8π²

B

ρL³/16π²

C

5 ρL³/16π²

D

3 ρL³/8π²

Question 48: Palette 48/197

Allen MCQ

One solid sphere A and another hollow sphere B are of same mass and same outer radii. Their moment of inertia about their diameters are respectively $I_{A}$ and $I_{B}$ such that (Where $d_{A}$ and $d_{B}$ are their densities)

A

$I_{A} = I_{B}$

B

$I_{A} > I_{B}$

C

$I_{A} < I_{B}$

D

$\frac{I_{A}}{I_{B}} = \frac{d_{A}}{d_{B}}$

Question 49: Palette 49/197

Allen MCQ

Three bodies have equal mass m. Body A is solid cylinder of radius R, body B is a square lamina of side R, and body C is a solid sphere of radius R. Which body has the smallest moment of inertia about an axis passing through their centre of mass and perpendicular to the plane (in case of lamina)

A

A

B

B

C

C

D

A and C both

Question 50: Palette 50/197

Allen MCQ

For the same total mass which of the following will have the largest moment of inertia about an axis passing through its centre of mass and perpendicular to the plane of the body

A

a disc of radius a

B

a ring of radius a

C

a square lamina of side 2a

D

four rods forming a square of side 2a

Question 51: Palette 51/197

Allen MCQ

Moment of inertia of a rectangular plate about an axis passing through P and perpendicular to the plate is I. Then moment of PQR about an axis perpendicular to the plane of the plate:

A

about $P = I / 2$

B

about $R = I / 2$

C

about $P > I / 2$

D

about $R > I / 2$

Question 52: Palette 52/197

Allen MCQ

Find the moment of inertia of a plate cut in shape of a right angled triangle of mass M, about an axis perpendicular to the plane of the plate and passing through the mid point of side AB. (Side AC = BC = a)

A

$\frac{Ma^2}{12}$

B

$\frac{Ma^2}{6}$

C

$\frac{Ma^2}{3}$

D

$\frac{2Ma^2}{3}$

Question 53: Palette 53/197

Allen MCQ

One quarter sector is cut from a uniform circular disc of radius R. This sector has mass M. It is made to rotate about a line perpendicular to its plane and passing through the center of the original disc. Its moment of inertia about the axis of rotation is

A

$\frac{1}{2} MR^2$

B

$\frac{1}{4} MR^2$

C

$\frac{1}{8} MR^2$

D

$\sqrt{2} MR^2$

Question 54: Palette 54/197

Allen MCQ

A rod non uniform mass distribution. The mass increases from left end (A) to right end (B). It is easiest to rotate the rod about an axis

A

Passing through A

B

Passing through B

C

Passing through x

D

Passing through y

Question 55: Palette 55/197

Allen MCQ

A semicircular lamina of mass m has radius r and centre C. Its moment of inertia about an axis through its centre of mass and perpendicular to its plane is- m

A

$\frac{1}{2} mr^2$

B

$\left(\frac{1}{2} -\frac{4}{\pi^2}\right)mr^2$

C

$\left(\frac{1}{2} -\frac{8}{9\pi^2}\right)mr^2$

D

$\left(\frac{1}{2} -\frac{16}{9\pi^2}\right)mr^2$

Question 56: Palette 56/197

Allen MCQ

A uniform rod of mass 2M is bent into four adjacent semicircles each of radius r, all lying in the same plane. The moment of inertia of the bent rod about an axis through one end A and perpendicular to plane of rod is :-

A

$22Mr^{2}$

B

$88Mr^{2}$

C

$44Mr^{2}$

D

$66Mr^{2}$

Question 57: Palette 57/197

Allen MCQ

Two identical rings each of mass m with their planes mutually perpendicular, radius R are welded at their point of contact O. If the system is free to rotate about an axis passing through the point P perpendicular to the plane of the paper the moment of inertia of the system about this axis is equal to

A

$6.5mR^{2}$
$$<br /> 1 2 m R ^ {2}<br /> $$

D

$11.5mR^{2}$

Question 58: Palette 58/197

Allen MCQ

A uniform disc has radius R. If a concentric smaller disc of radius x is removed from the bigger disc then radius of gyration of the remaining disc about an axis passing through centre and perpendicular to plane of disc varies with x, according to the graph :- Radius of gyration (K)

A

B

Radius of gyration (K)

Radius of gyration (K)

C

D

Radius of gyration (K)

Question 59: Palette 59/197

Allen MCQ

A weightless rod is acted on by upward parallel forces of 2N and 4N at ends A and B respectively. The total length of the rod is AB = 3m. To keep the rod in equilibrium a force of 6N should act in the following manner:

A

Downwards at any point between $A$ and $B$ .

B

Downwards at mid point of $AB$ .

C

Downwards at a point C such that AC = 1m.

D

Downwards at a point D such that BD = 1m.

Question 60: Palette 60/197

Allen MCQ

A right triangular plate ABC of mass m is free to rotate in the vertical plane about a fixed horizontal axis through A. It is supported by a string such that the side AB is horizontal. The reaction at the support A in equilibrium is:

A

$\frac{mg}{3}$

B

$\frac{2mg}{3}$

C

$\frac{mg}{2}$

D

mg

Question 61: Palette 61/197

Allen MCQ

A uniform rod of length l is placed symmetrically on two walls as shown in figure. The rod is in equilibrium. If $N_{1}$ and $N_{2}$ are the normal forces exerted by the walls on the rod then :-

A

$N_{1} > N_{2}$

B

$N_{1} < N_{2}$

C

$N_{1} = N_{2}$

D

$N_{1}$ and $N_{2}$ would be in the vertical directions

Question 62: Palette 62/197

Allen MCQ

A vertical rectangular door with its centre of gravity at O (see figure) is fixed on two hinges A and B along one vertical length side of the door. The entire weight of the door is supported by the hinge A. Then the free body force diagram for the door (the arrows indicate the direction of the forces) is:

A

B

C

D

Question 63: Palette 63/197

Allen MCQ

Same number of books are placed in four book cases as shown. Which bookcase is most likely to topple forward if pulled a little at the top towards right:

A

B

C

D

Question 64: Palette 64/197

Allen MCQ

A homogeneous cubical brick lies motionless on a rough inclined surface. The half of the brick which applies greater pressure on the plane is :

A

left half

B

right half

C

both applies equal pressure

D

the answer depend upon coefficient of friction

Question 65: Palette 65/197

Allen MCQ

A cubical block of side L rests on a rough horizontal surface with coefficient of friction $\mu$ . A horizontal force F is applied on the block as shown. If the coefficient of friction is sufficiently high so that the block does not slide before toppling, the minimum force required to topple the block is $\tau = I\alpha$ Torque

A

Infinitesimal

B

mg/4

C

mg/2

D

$mg(1 - m)$

Question 66: Palette 66/197

Allen MCQ

A solid cone hangs from a frictionless pivot at the origin O, as shown. If $\hat{i}, \hat{j}$ and $\hat{k}$ are unit vectors, and a, b, and c are positive constants, which of the following forces F applied to the rim of the cone at a point P results in a torque $\tau$ on the cone with a negative component $\tau_{z}$ ?

A

$F = a\hat{k}, P$ is $(0, b, -c)$

B

$F = -a\hat{k}, P$ is $(0, -b, -c)$

C

$F = a\hat{j},P$ is $(-b,0, - c)$

D

None

Question 67: Palette 67/197

Allen MCQ

A sphere of mass m and radius R is kept on a rough horizontal surface and pulled with a horizontal force $F = \alpha t$ where $\alpha$ is a constant and t is time. The coefficient of static friction is $\mu_{s}$ and coefficient of kinetic friction is $\mu_{k}$ . Which of the following graph correctly shows the variation of acceleration of sphere with time?

A

B

C

D

Question 68: Palette 68/197

Allen MCQ

A homogeneous disc with a radius 0.2 m and mass 5 kg rotates around an axis passing through its centre. The angular velocity of the rotation of the disc as a function of time is given by the formula $\omega = 2 + 6t$ . The tangential force applied to the rim of the disc is

A

1 N

B

2 N

C

3 N

D

4 N

Question 69: Palette 69/197

Allen MCQ

A thin uniform hemispherical bowl of mass m and radius R is lying on a smooth horizontal surface. A horizontal force F is now applied perpendicular to the rim of the bowl (see figure). The instantaneous angular acceleration of the bowl will be :-

A

$\frac{20}{3}\frac{F}{MR}$

B

$\frac{10}{3}\frac{F}{MR}$

C

$\frac {4 0}{3} \frac {F}{M R}$

D

$\frac{6}{5}\frac{F}{MR}$

Question 70: Palette 70/197

Allen MCQ

A uniform rod AB of weight 100 N rests on a rough peg at C and a force F acts at A as shown in figure. BC = CM, M is the midpoint of AB and $\tan\alpha = 4/3$ . The minimum coefficient of friction at C is

A

$9 / 8$

B

$3 / 2$

C

$8 / 7$

D

$4 / 3$

Question 71: Palette 71/197

Allen MCQ

A particle of mass 2 kg located at the position $(\hat{i} + \hat{j})$ m has a velocity $2 (+\hat{i} - \hat{j} + \hat{k})$ m/s. Its angular momentum about z-axis in $kg-m^{2}/s$ is:

A

zero

B

+8

C

12

D

-8

Question 72: Palette 72/197

Allen MCQ

A ball of mass m moving with velocity v, collide with the wall elastically as shown in the figure. After impact the change in angular momentum about P is:

A

$2mvd$

B

$2mvd\cos\theta$

C

$2mvd\sin\theta$

D

zero

Question 73: Palette 73/197

Allen MCQ

Two uniform spheres of mass M have radii R and 2R. Each sphere is rotating about a fixed axis through a diameter. The rotational kinetic energies of the spheres are identical. What is the ratio of the magnitude of the angular momenta of these spheres? That is, $\frac{L_{2R}}{L_{R}} =$

A

4

B

$2\sqrt{2}$

C

2

D

$\sqrt{2}$

Question 74: Palette 74/197

Allen MCQ

A particle of mass m moves with a constant velocity. Which of the following statements is not correct about its angular momentum about point O :

A

it is zero when it is at A and moving along OA

B

is the same at all points along the line DE

C

is of the same magnitude but oppositely directed at B and D

D

increases as it moves along the line BC

Question 75: Palette 75/197

Allen MCQ

A projectile of mass 'm' is thrown from the origin O with a velocity $v_{0}$ at an angle $\theta$ . Identify the CORRECT statement :-

A

Angular momentum of the particle about O is always zero

B

Angular momentum of the particle about O increases with time.

C

A constant torque acts on the particle about O in the clockwise sense.

D

Angular momentum of the particle is least when it is at the highest position.

Question 76: Palette 76/197

Allen MCQ

A particle of mass m is projected from point O with a horizontal velocity 'u' at time t = 0. What is its angular momentum relative to point O as a function of time?

A

$-\frac{1}{2} mgut^{2}\hat{k}$

B

$\frac{1}{2} mgut^{2}\hat{k}$

C

$-mgut^{2}\hat{k}$

D

$+mgut^{2}\hat{k}$

Question 77: Palette 77/197

Allen MCQ

An equilateral triangle ABC formed from a uniform wire has two small identical beads initially located at A. The triangle is set rotating about the vertical axis AO. Then the beads are released from rest simultaneously and allowed to slide down, one along AB and the other along AC as shown. Neglecting frictional effects, the quantities that are conserved as the beads slide down, are

A

Angular velocity and total energy (kinetic and potential)

B

Total angular momentum and total energy

C

Angular velocity and moment of inertia about the axis of rotation.

D

Total angular momentum and moment of inertia about the axis of rotation.

Question 78: Palette 78/197

Allen MCQ

A circular platform is free to rotate in a horizontal plane about a vertical axis passing through its centre. A tortoise is sitting at the edge of the platform. Now, the platform is given an angular velocity $\omega_{0}$ . When the tortoise moves along a chord of the platform with a constant velocity (with respect to the platform), the angular velocity of the platform $\omega(t)$ . will vary with time t as

A

B

C

D

Question 79: Palette 79/197

Allen MCQ

Two identical discs each of mass $M(=4m)$ and radius R are rotating in opposite sense with equal angular speed $\omega_{0}$ about vertical axes passing through their centres, (as shown in figure). A person of mass m sitting on circumference of disc A jumps with a tangential relative velocity u (after jumping) w.r.t one rotating disc (A) and lands on other disc (B) also tangential. Now the second disc (B) comes to a stop. Find the relative velocity u.

A

$u = \frac {\omega_ {0} R}{2}$

B

$u = \omega_0 R$

C

$u = \frac{3\omega_0 R}{2}$`

D

$u = 2 \omega_ {0} R$

Question 80: Palette 80/197

Allen MCQ

A spinning ice skater can increase his rate of rotation by bringing his arms and free leg closer to his body. How does this procedure affect the skater's angular momentum and kinetic energy and what is the work done by the skater?

A

angular momentum remains the same while kinetic energy increases and work done is positive.

B

angular momentum remains the same while kinetic energy decreases and work done is negative.

C

both angular momentum and kinetic energy remain the same and work done is zero.

D

angular momentum increases while kinetic energy remains the same and work done may be positive or negative.

Question 81: Palette 81/197

Allen MCQ

A child with mass m is standing at the edge of a disc with moment of inertia I, radius R, and initial angular velocity $\omega$ . See figure given below. The child jumps off the edge of the disc with tangential velocity v with respect to the ground. The new angular velocity of the disc is

A

$\sqrt{\frac{I\omega^{2}-mv^{2}}{I}}$

B

$\sqrt{\frac{(I+mR^{2})\omega^{2}-mv^{2}}{I}}$

C

$\frac{I\omega-mvR}{I}$

D

$\frac{(I+mR^{2})\omega-mvR}{I}$

Question 82: Palette 82/197

Allen MCQ

A rod of mass M and length $\ell$ stands along z-axis. Its lower end is hinged at the centre of a disc of same mass M and radius R. The whole arrangement is rotating freely about z-axis with an angular velocity $\omega_{0}$ . The rod falls on disc & rotates with disc. Find the angular speed of the arrangement.

A

$\frac{\omega_{0}R^{2}}{\ell^{2}+R^{2}}$

B

$\frac{3\omega_{0}R^{2}}{2\ell^{2}+3R^{2}}$

C

$\frac{6\omega_{0}R^{2}}{\ell^{2}+6R^{2}}$

D

$\frac{6\omega_{0}R}{\ell+6R}$

Question 83: Palette 83/197

Allen MCQ

A child is standing on the edge of a merry-go-round that has the shape of a disk, as shown in the figure above. The mass of the child is 40 kilograms. The merry-go-round has a mass of 200 kilograms and a radius of 2.5 meters, and it is rotating with an angular velocity of $\omega = 2.0$ radians per second. The child then walks slowly toward the center of the merry-go-round. What will be the final angular velocity of the merry-go-round when the child reaches the center? (The size of the child can be neglected)

A

2.0 rad/s

B

2.2 rad/s

C

2.4 rad/s

D

2.8 rad/s

Question 84: Palette 84/197

Allen MCQ

A uniform disk turns at 2.4 rev/s around a frictionless axis. A nonrotating rod, of the same mass as the disk and length equal to the disk's diameter is dropped onto the freely spinning disk figure. They then both turn around the axis with their centres superposed. What as the angular frequency in rev/s of the combination.

A

1.2 rev/s

B

2.0 rev/s

C

1.44 rev/s

D

0.96 rev/s

Question 85: Palette 85/197

Allen MCQ

A disc is rolling without slipping with angular velocity $\omega$ . P and Q are two points equidistant from the centre C. The order of magnitude of velocity is :-

A

$v_{Q} > v_{C} > v_{p}$

B

$v_{P} > v_{C} > v_{Q}$

C

$v_{P} = v_{C}, v_{Q} = v_{C} / 2$

D

$v_{P} < v_{C} > v_{Q}$

Question 86: Palette 86/197

Allen MCQ

In the following figure, a sphere of radius 3 m rolls on a plank. The accelerations of the sphere and the plank are indicated. The value of $\alpha$ is

A

3 rad/s $^{2}$

B

$6 \, rad/s^{2}$

C

$3 \, rad/s^{2}$ (opposite to the direction shown in figure)

D

$1\mathrm{rad} / s^{2}$

Question 87: Palette 87/197

Allen MCQ

A disc of radius R is rolling purely on a flat horizontal surface, with a constant angular velocity. The angle between the velocity and acceleration vectors of point P is

A

zero

B

$45^{\circ}$

C

$135^{\circ}$

D

$\tan^{-1}(1/2)$

Question 88: Palette 88/197

Allen MCQ

A cylinder is placed on a rough plank which in turn is placed on a smooth surface. The plank is pulled with a constant force F. The friction force can be given by which of the following diagrams

A

B

C

D

cannot be interpreted

Question 89: Palette 89/197

Allen MCQ

A body kept on a smooth horizontal surface is pulled by a constant horizontal force applied at the top point of the body. If the body rolls purely on the surface, its shape can be :

A

thin pipe

B

uniform cylinder

C

uniform sphere

D

thin spherical shell

Question 90: Palette 90/197

Allen MCQ

A hollow sphere of radius R and mass m is fully filled with non viscous liquid of mass m. It is rolled down a horizontal plane such that its centre of mass moves with a velocity v. If it purely rolls

A

Kinetic energy of the sphere is $\frac{5}{6}mv^{2}$

B

Kinetic energy of the sphere is $\frac{4}{5}mv^{2}$

C

Angular momentum of the sphere about a fixed point on ground is $\frac{8}{3}mvR$

D

Angular momentum of the sphere about a fixed point on ground is $\frac{14}{5}mvR$

Question 91: Palette 91/197

Allen MCQ

A sphere of mass M and radius R is attached by a light rod of length $\ell$ to a point P. The sphere rolls without slipping on a circular track as shown. It is released from the horizontal position. The angular momentum of the system about P when the rod becomes vertical is :

A

$M\sqrt{\frac{10}{7}gl}[l + R]$

B

$M\sqrt{\frac{10}{7}gl}\left[l + \frac{2}{5}R\right]$

C

$M \sqrt{\frac{10}{7} gl} \left[ l + \frac{7}{5} R \right]$

D

$M\sqrt{\frac{10}{7}gl}\left[l - \frac{2}{5}R\right]$

Question 92: Palette 92/197

Allen MCQ

A thin table cloth covers a horizontal table and a uniform body of round shape lies on top of it. The table cloth is pulled from under the body, and friction causes the body to slide and rotate. What is the body's final motion on the table? (Assume that the table is so large that the body does not fall off it.)

A

Body will finally roll towards left

B

Body will finally roll towards right

C

Body will finally come to rest

D

Any of the above is possible depending on shape of body

Question 93: Palette 93/197

Allen MCQ

A ball rolls down an inclined plane as shown in figure. The ball is first released from rest from P and then later from Q. Which of the following statement is/are correct? (i) The ball takes twice as much time to roll from Q to O as it does to roll from P to O. (ii) The acceleration of the ball at Q is twice as large as the acceleration at P. (iii) The ball has twice as much K.E. at O when rolling from Q as it does when rolling from P.

A

i, ii only

B

ii, iii only

C

i only

D

iii only

Question 94: Palette 94/197

Allen MCQ

A metal ball of mass m is put at the point A of a loop tract and the vertical distance of A from the lower most point of tract is n times the radius R of the circular part. The linear velocity of ball when it rolls to the point B to a height R in the circular tract will be

A

$\sqrt{\frac{5g(n - 1)R}{7}}$

B

$\sqrt{\frac{10g(n - 1)R}{7}}$

C

$\sqrt{\frac{5g(n+1)R}{7}}$

D

$\sqrt{\frac{10g(n+1)R}{7}}$

Question 95: Palette 95/197

Allen MCQ

A ring of mass m is thrown in horizontal direction on a rough horizontal surface with a speed $v_{0}$ . The speed of the ring, when it starts pure rolling, is –

A

$\frac{v_{0}}{2}$

B

$\frac{v_{0}}{3}$

C

$\frac{2v_{0}}{3}$

D

$\frac{3v_{0}}{3}$

Question 96: Palette 96/197

Allen MCQ

A solid sphere of mass M and radius r slips on a rough horizontal plane. At some instant it has translational velocity $v_{0}$ and rotational velocity about the centre $v_{0}/2r$ . The translational velocity after the sphere starts pure rolling

A

$6v_{0}/7$ in forward direction

B

$6v_{0}/7$ in backward direction

C

$7v_{0}/6$ in forward direction

D

$6v_{0}/6$ in backward direction

Question 97: Palette 97/197

Allen MCQ

A hollow sphere is moving with linear velocity and angular velocity as shown in figure. The time after which pure rolling starts is :

A

$\frac{2v}{3\mu g}$

B

$\frac{4v}{7\mu g}$

C

$\frac{4v}{5\mu g}$

D

$\frac{5v}{4\mu g}$

Question 98: Palette 98/197

Allen MCQ

A uniform circular disc placed on a rough horizontal surface has initially a velocity $v_{0}$ and an angular velocity $\omega_{0}$ as shown in the figure. The disc comes to rest after moving some distance in the direction of motion. Then $\frac{v_{0}}{r\omega_{0}}$ is

A

$\frac {1}{2}$

B

1

C

$\frac {3}{2}$

D

2

Question 99: Palette 99/197

Allen MCQ

A solid sphere is under pure rolling on a rough fixed incline plane of angle $\theta$ . Choose the correct options if only contact forces and gravity are acting :

A

Frictional force will be down the incline if sphere rolls up the incline and it will be up the incline if sphere rolls down the incline.

B

Frictional force will be down the incline whether spheres rolls up the incline or down the incline

C

Frictional force and acceleration of the body will increase with the increase in angle of incline plane

D

Velocity and acceleration of the point of contact of sphere with incline will be zero during the motion.

Question 100: Palette 100/197

Allen MCQ

A solid uniform disk of mass m rolls without slipping down a fixed inclined plane with an acceleration a. The frictional force on the disk due to surface of the plane is:

A

2 ma

B

3/2 ma

C

ma

D

1/2 ma

Question 101: Palette 101/197

Allen MCQ

The cylinder shown, with mass M and radius R, has a radially dependent density. The cylinder starts from rest and rolls without slipping down an inclined plane of height H. At the bottom of the plane its translational speed is $(8gH/7)^{1/2}$ . Which of the following is the rotational inertia of the cylinder?

A

$\frac {1}{2} M R ^ {2}$

B

$\frac{3}{4} MR^2$

C

$\frac{7}{8} MR^2$

D

$\mathrm{MR}^2$

Question 102: Palette 102/197

Allen MCQ

A uniform disk of mass m and radius R rolls without slipping down an incline plane of length l and inclination $\theta$ . Initially the disk was at rest at the top of the incline plane. Its angular momentum about the point of contact with the inclined plane when it reaches the bottom will be equal to:

A

$\sqrt{4m^{2}R^{2}gl\sin\theta}$

B

$\sqrt{3m^{2}R^{2}gl\sin\theta}$

C

$\sqrt{\frac{m^{2}R^{2}gl\sin\theta}{4}}$

D

zero

Question 103: Palette 103/197

Allen MCQ

A cylinder rolls up an inclined plane, reaches some height, and then rolls down (without slipping throughout these motions). The directions of the frictional force acting on the cylinder are

A

Up the incline while ascending and down the incline while descending

B

Up the incline while ascending as well as descending

C

down the incline while ascending and up the incline while descending

D

down the incline while ascending as well as descending

Question 104: Palette 104/197

Allen MCQ

A plank of mass M is placed over smooth inclined plane and a sphere is also placed over the plank. Friction is sufficient between sphere and plank. If plank and sphere are released from rest, the frictional force on sphere is:

A

up the plane

B

down the plane

C

horizontal

D

zero

Question 105: Palette 105/197

Allen MCQ

A force F is applied to a dumbbell for a time interval, t, first as in (i) and then as in (ii). In which case does the dumbbell acquire the greater centre-of-mass speed?

A

(i)

B

(ii)

C

there is no difference

D

the answer depends on the rotational inertia of the dumbbell

Question 106: Palette 106/197

Allen MCQ

A uniform rod of length l and mass M rotating about a fixed vertical axis on a smooth horizontal table. It elastically strikes a particle placed at a distance l/3 from its axis and stops. Mass of the particle is

A

$3M$

B

$\frac{3M}{4}$

C

$\frac{3M}{2}$

D

$\frac{4M}{3}$

Question 107: Palette 107/197

Allen MCQ

A mass m is moving at speed v perpendicular to a rod of length d and mass M = 6m which pivots around a frictionless axle running through its centre. It strikes and sticks to the end of the rod. The moment of inertia of the rod about its centre is $Md^{2}/12$ . Then the angular speed of the system right after the collision is

A

2v/d

B

2v/(3d)

C

v/d

D

3v/(2d)

Question 108: Palette 108/197

Allen MCQ

Two particles each of mass M are connected by a massless rod of length l. The rod is lying on the smooth surface. If one of the particle is given an impulse MV as shown in the figure then angular velocity of the rod would be

A

v/l

B

$2v / l$

C

v/2l

D

None

Question 109: Palette 109/197

Allen MCQ

A uniform rod AB of mass m and length l is at rest on a smooth horizontal surface. An impulse J is applied to the end B, perpendicular to the rod in the horizontal direction. Speed of particle P at a distance $\frac{l}{6}$ from the centre towards A of the rod after time $t = \frac{\pi m l}{12J}$ is

A

$2 \frac{J}{m}$

B

$\frac{J}{\sqrt{2}m}$

C

$\frac{J}{m}$

D

$\sqrt{2}\frac{J}{m}$

Question 110: Palette 110/197

Allen MCQ

A sphere is placed rotating with its centre initially at rest in a corner as shown in figure (a) and (b). Coefficient of friction between all surfaces and the sphere is 1/3. Find the ratio of the frictional force $\frac{f_a}{f_b}$ by ground in situations (a) and (b).
(a)

(b)

A

1

B

$\frac{9}{10}$

C

$\frac{10}{9}$

D

none

Question 111: Palette 111/197

Allen MCQ

A yo-yo is resting on a perfectly rough horizontal table. Forces $F_{1}$ , $F_{2}$ and $F_{3}$ are applied separately as shown. The correct statement is :-

A

when $F_{3}$ is applied the centre of mass will move to the right.

B

when $F_{2}$ is applied the centre of mass will move to the left.

C

when $F_{1}$ is applied the centre of mass will move to the right.

D

when $F_{2}$ is applied the centre of mass will move to the right.

Question 112: Palette 112/197

Allen MCQ

One ice skater of mass m moves with speed 2v to the right, while another of the same mass m moves with speed v toward the left, as shown in figure I. Their paths are separated by a distance b. At t = 0, when they are both at x = 0, they grasp a pole of length b and negligible mass. For t > 0, consider the system as a rigid body of two masses m separated by distance b, as shown in figure II. Which of the following is the correct formula for the motion after t = 0 of the skater initially at y = b/2?

A

$x = 2vt, y = b / 2$

B

$x = vt + 0.5b\sin (3vt / b),y = 0.5b\cos (3vt / b)$

C

$x = 0.5vt + 0.5b \sin(3vt/b), y = 0.5b \cos(3vt/b)$

D

$x = 0.5vt + 0.5b \sin(6vt/b), y = 0.5b \cos(6vt/b)$

Question 113: Palette 113/197

Allen MSQ

A uniform rod of mass M = 2 kg and length L is suspended by two smooth hinges 1 and 2 as shown in the figure. A force F = 4N is applied downward at a distance from the hinge 2. Due to the application of force F, the hinge 2 breaks. At this instant, applied force F is also removed. The rod starts to rotate downward about hinge 1. $(g = 10 \, m/s^{2})$ . Choose the correct statement(s) :-

A

The reaction at hinge 1, before hinge 2 breaks, is 11 N

B

The reaction at hinge 1, before hinge 2 breaks, is 24 N

C

The reaction at hinge 1, just after breaking of hinge 2 is 5N

D

The reaction at hinge 1, just after breaking of hinge 2 is 8N

Question 114: Palette 114/197

Allen MSQ

A thin ring is free to rotate in the vertical plane about O. Ring is released from rest at $\theta = 0$ , when ring rotates through angle $\theta$ :

A

Angular acceleration of ring is $\frac{g}{2}\frac{\cos\theta}{R}$

B

Centripetal acceleration of ring is $2g \sin\theta$

C

Net force by hinge is $\frac{mg}{2}\sqrt{15\sin^{2}\theta+1}$

D

Net force by hinge is $mg\sqrt{4\sin^{2}\theta+\frac{3}{4}\cos^{2}\theta}$

Question 115: Palette 115/197

Allen MSQ

Two particles each of mass m are attached at end points of a massless rod AB of length $\ell$ . Rod is hinged at point C as shown. Rod is released from rest from horizontal position. At the instant when rod reaches its vertical position as shown, which of the following is/are CORRECT :-

A

Speed of the particle at B is thrice the speed of particle at A

B

Net force on particle B is $\frac{6mg}{5}$

C

Angular acceleration of the system is zero

D

Both $x \& y$ components of hinge force are nonzero

Question 116: Palette 116/197

Allen MSQ

A rod is connected with the (rigid body) disc at centre of disc point 'P'. The disc can freely rotate about point P where it is hinged with rod and rod can rotate freely about fixed hinge Q. Now whole arrangement is released from horizontal position as shown in the diagram.

A

Initial angular acceleration of (rigid body) disc is zero.

B

Initial angular acceleration of (rigid body) disc is $\frac{9g}{8\ell}$ .

C

Angular speed of (rigid body) disc when rod becomes vertical is zero.

D

Angular speed of (rigid body) disc when rod becomes vertical is $\frac{3}{2}\sqrt{\frac{g}{\ell}}$ rad/s.

Question 117: Palette 117/197

Allen MSQ

ABCD is a square plate with centre O. The moments of inertia of the plate about the perpendicular axis through O is I and about the axes 1, 2, 3 and 4 are $I_{1}$ , $I_{2}$ , $I_{3}$ and $I_{4}$ respectively. It follows that:

A

$I_{2} = I_{3}$

B

$I = I_{1} + I_{4}$

C

$I = I_{2} + I_{4}$

D

$I_{1} = I_{3}$

Question 118: Palette 118/197

Allen MSQ

A block with a square base measuring 'a' × 'a' and height h, is placed on an inclined plane. The coefficient of friction is μ. The angle of inclination (θ) of the plane is gradually increased. The block will

A

topple before sliding if $\mu > \frac{a}{h}$

B

topple before sliding if $\mu < \frac{a}{h}$

C

slide before toppling if $\mu > \frac{a}{h}$

D

slide before toppling if $\mu < \frac{a}{h}$

Question 119: Palette 119/197

Allen MSQ

A particle falls freely near the surface of the earth. Consider a fixed point O (not vertically below the particle) on the ground.

A

Angular momentum of the particle about O is increasing.

B

Torque of the gravitational force on the particle about O is decreasing.

C

The moment of inertia of the particle about O is decreasing.

D

The angular velocity of the particle about O is increasing.

Question 120: Palette 120/197

Allen MSQ

A small ball of mass m suspended from the ceiling at a point O by a thread of length $\ell$ moves along a horizontal circle with a constant angular velocity $\omega$ .

A

angular momentum is constant about O

B

angular momentum is constant about C

C

vertical component of angular momentum about O is constant

D

Magnitude of angular momentum about O is constant

Question 121: Palette 121/197

Allen MSQ

A block of mass m moves on a horizontal rough surface with initial velocity v. The height of the centre of mass of the block is h from the surface. Consider a point A on the surface.

A

angular momentum about A is mvh initially

B

the velocity of the block decreases at time passes.

C

torque of the forces acting on block is zero about A

D

angular momentum is not conserved about A.

Question 122: Palette 122/197

Allen MSQ

A satellite in outer-space is at an initial rotation speed of $\omega = 4$ rad/sec at t = 0. As it deploys its solar panels, it changes its rotational inertia as shown in graph below. The correct plot of magnitude of angular momentum and angular velocity as functions of time :

A

B

C

D

Question 123: Palette 123/197

Allen MSQ

In making a science-fiction movie about an asteroid that collides with Earth, a studio constructs a model Earth pivoted at a support fixed at the south Pole and spinning rapidly about the (vertical) south-to-north axis with angular speed $\omega$ pointing out of the North Pole. A Model asteroid approaches horizontally and strikes a grazing blow with the model Earth just at the North Pole. Describe the new motion of the Model Earth.

A

The earth now rotates about an axis which is initially in YZ plane but rotates about the Z axis.

B

The earth now rotates about an axis which is initially in XZ plane but rotates about the Z axis.

C

The angular velocity of earth in Z direction remains the same

D

The angular momentum of earth in Z direction remains the same.

Question 124: Palette 124/197

Allen MSQ

A circular disc of radius R rolls without slipping on a rough horizontal surface. At the instant shown its linear velocity is V, linear acceleration a, angular velocity $\omega$ and angular acceleration a. Four points A, B, C and D lie on its circumference such that the diameter AC is vertical and BD horizontal then choose the correct option(s).

A

$V_{B} = \sqrt{V^{2} + (R\omega)^{2}}$

B

$V_{C} = V + R\omega$

C

$a_{A} = \sqrt{(a - R\alpha)^{2} + (\omega^{2}R)^{2}}$

D

$a_{D} = \sqrt{(a + \omega^{2}R)^{2} + (R\alpha)^{2}}$

Question 125: Palette 125/197

Allen MSQ

The disc of radius r is confined to roll without slipping at A and B. If the plates have the velocities shown, then

A

linear velocity $v_{0} = v$

B

angular velocity of disc is $\frac{3v}{2r}$

C

angular velocity of disc is $\frac{2v}{r}$

D

None of these

Question 126: Palette 126/197

Allen MSQ

A massless rod has a massless hollow sphere attached to it. This sphere can be fully filled either with a liquid (non viscous) or with a solid (rigidly fitted into sphere) of same mass. System is released from rest from initial position (as shown). When it reaches final position which of the following is/are true for the system.

A

Kinetic energy in case of liquid will be more than in case of solid.

B

Velocity of centre (V) in case of liquid will be more than in case of solid.

C

About C angular momentum in case of liquid will be more than in case of solid.

D

About C angular momentum in case of liquid will be less than in case of solid.

Question 127: Palette 127/197

Allen MSQ

A disc of radius a is rigidly attached at its circumference by two rods each of length 3a, two rods are always perpendicular to each other. The combination is suspended vertically from the free end of one rod (as shown in figure). It is swinging in the plane of the disc such that the centre of disc have velocity v. If mass of disc and each rod is m.

A

Kinetic energy of the disc $\frac{1}{2} mV^2 +\frac{1}{2}\left(\frac{ma^2}{2}\right)\left(\frac{V}{4a}\right)^2$

B

Kinetic energy of the rod 2 is $\frac{1}{2} mV^2 +\frac{1}{2}\times \left(\frac{m(3a)^2}{12}\right)\times \left(\frac{V}{4a}\right)^2$

C

Kinetic energy of the rod 1 is $\frac{1}{2} \times \left( \frac{m(3a)^2}{3} \right) \times \left( \frac{V}{4a} \right)^2$

D

Kinetic energy of the system of (disc and both rod) is
$$<br /> \frac {1}{2} \left[ \left(\frac {m (3 a) ^ {2}}{3}\right) + \left(\frac {m a ^ {2}}{2} + m (4 a) ^ {2}\right) + \left(\frac {m (3 a) ^ {2}}{1 2} + m \frac {8 9}{4} a ^ {2}\right) \right] \left(\frac {V}{4 a}\right) ^ {2}<br /> $$

Question 128: Palette 128/197

Allen MSQ

A disc of mass ‘m’ and radius ‘R’ is rotating with angular speed $\omega_{0}$ and speed zero as shown and then placed on a rough surface (coefficient of friction= $\mu$ ) and left. Then

A

angular momentum of disc is conserved about point Q.

B

angular momentum of disc is conserved about point P on the ground.

C

initially the force of friction is $\mu mg$ leftward.

D

initially the force of friction is $\mu$ mg rightward.

Question 129: Palette 129/197

Allen MSQ

A billiard ball initially at rest is given a sharp blow by a cue stick. The impulse is horizontal and is applied at a distance 2R/3 below the centreline, as shown in figure. The initial speed of the ball is $v_{0}$ , and the coefficient of kinetic friction is $\mu_{k}$ .

A

Initially kinetic friction acts in $-\hat{i}$ direction

B

Initially kinetic friction acts in $\hat{i}$ direction.

C

Ball instantaneously starts pure rolling.

D

Initial angular velocity of ball is $\frac{5v_{0}}{3R}$

Question 130: Palette 130/197

Allen MSQ

A solid sphere of mass 10 kg is placed on a rough horizontal surface with coefficient of friction $\mu = 0.1$ . If a force of 28 N is applied at the top of the sphere then,

A

friction force will be 10 N in forward direction.

B

friction force will be less than 10N in backward direction.

C

Acceleration of centre of mass will be $\frac{29}{7}$ m/s $^{2}$ .

D

Acceleration of centre of mass will be $3.8m/s^{2}$ .

Question 131: Palette 131/197

Allen MSQ

If a cylinder is rolling down the incline with sliding.

A

after some time it may start pure rolling

B

after sometime it will start pure rolling

C

it may be possible that it will never start pure rolling

D

none of these

Question 132: Palette 132/197

Allen MSQ

A plank with a uniform sphere placed on it rests on a smooth horizontal plane. Plank is pulled to right by a constant force F. If sphere does not slip over the plank. Which of the following is correct.

A

Acceleration of the centre of sphere is less than that of the plank.

B

Work done by friction acting on the sphere is equal to its total kinetic energy.

C

Total kinetic energy of the system is equal to work done by the force F.

D

None of the above

Question 133: Palette 133/197

Allen MSQ

A uniform rod AB of length $\ell$ and mass M hangs from point A at which it is freely hinged in a car moving with velocity $v_{0}$ . The rod can rotate in vertical plane about the axis at A. If the car suddenly stops,

A

The angular speed $\omega$ with which the rod starts rotating is $\frac{3v_{0}}{2\ell}$ .

B

The minimum value of $v_{0}$ so that the rod completes the rotation $\sqrt{\frac{8}{3} g^{\ell}}$ .

C

Loss of energy during the process $\frac{1}{8} M v_0^2$ .

D

There is no loss of energy.

Question 134: Palette 134/197

Allen MCQ

Comprehension Passage: PARAGRAPH FOR QUESTION NO. 106 TO 108

A rod of length $\ell$ and mass m is hinged at A. It is released from horizontal position from rest. When it comes to position AB, and makes and angle of $30^{\circ}$ with vertical Then the angular velocity in this position is:-

A

$\left(\frac{3\sqrt{3}g}{2\ell}\right)^{1/2}$

B

$\left(\frac{3g}{2\ell}\right)^{1/2}$

C

$\left(\frac{9g}{2\ell}\right)^{1/2}$

D

$\left(\frac{3\sqrt{3}g}{4\ell}\right)^{1/2}$

Question 135: Palette 135/197

Allen MCQ

Comprehension Passage: PARAGRAPH FOR QUESTION NO. 106 TO 108

A rod of length $\ell$ and mass m is hinged at A. It is released from horizontal position from rest. When it comes to position AB, and makes and angle of $30^{\circ}$ with vertical Force applied by the hinge on the rod along the rod is :-

A

$\frac{3mg}{4}$

B

$\frac{5\sqrt{3}mg}{4}$

C

$\frac{3\sqrt{3}mg}{4}$

D

$\frac{\sqrt{3}mg}{2}$

Question 136: Palette 136/197

Allen MCQ

Comprehension Passage: PARAGRAPH FOR QUESTION NO. 106 TO 108

A rod of length $\ell$ and mass m is hinged at A. It is released from horizontal position from rest. When it comes to position AB, and makes and angle of $30^{\circ}$ with vertical A ladder of length L is slipping with its ends against a vertical wall and a horizontal floor. At a certain moment, the speed of the end in contact with the horizontal floor is v and the ladder makes an angle $\alpha = 30^{\circ}$ with the horizontal. Then the speed of the ladder's center must be :-

A

$\frac{2v}{\sqrt{3}}$

B

v/2

C

v

D

None

Question 137: Palette 137/197

Allen MCQ

Comprehension Passage: PARAGRAPH FOR QUESTION NO. 109 AND 110

A small sphere of mass 1 kg is rolling without slipping on a rough stationary base with linear speed $v = \sqrt{\frac{200}{7}}$ m/s. It leaves the inclined plane at point C. Find its linear speed at point C:-

A

$\sqrt{\frac{100}{7}} m / s$

B

$\sqrt{\frac{50}{7}} m / s$

C

$\sqrt{\frac{100}{35}} m / s$

D

$\sqrt{\frac{200}{35}} m / s$

Question 138: Palette 138/197

Allen MCQ

Comprehension Passage: PARAGRAPH FOR QUESTION NO. 109 AND 110

A small sphere of mass 1 kg is rolling without slipping on a rough stationary base with linear speed $v = \sqrt{\frac{200}{7}}$ m/s. It leaves the inclined plane at point C. Find ratio of rotational and translational kinetic energy of the sphere when it strikes the ground after leaving from point C:-

A

$\frac{2}{5}$

B

$\frac{2}{3}$

C

$\frac{1}{6}$

D

$\frac{1}{2}$

Question 139: Palette 139/197

Allen MCQ

Comprehension Passage: PARAGRAPH FOR QUESTION NO. 111 AND 112

In the treatment of moments of inertia, introductory textbooks often present two theorems, generally called the parallel axis theorem and the perpendicular axis theorem. There is another theorem of this same genre, which is not usually included, but which is interesting and useful. It is $$ I _ {x} + I _ {y} + I _ {z} = 2 \sum_ {i} m _ {i} r _ {i} ^ {2} $$ Here, $I_{x}$ , $I_{y}$ and $I_{z}$ are the moments of inertia about three mutually perpendicular intersecting axes, $m_{1}$ is the mass of the $i^{th}$ particle and $r_{i}$ is the distance from the intersection. The proof is simple: Taking the three axes as coordinate axes, have : $$ \begin{array}{r l} & I _ {x} + I _ {y} + I _ {z} \\ & = \sum_ {i} m _ {i} (y _ {i} ^ {2} + z _ {i} ^ {2}) + \sum_ {i} m _ {i} (z _ {i} ^ {2} + x _ {i} ^ {2}) + \sum_ {i} m _ {i} (x _ {i} ^ {2} + y _ {i} ^ {2}) \\ & = 2 \sum_ {i} m _ {i} (x _ {i} ^ {2} + y _ {i} ^ {2} + z _ {i} ^ {2}) = 2 \sum_ {i} m _ {i} r _ {i} ^ {2} \end{array} $$ One important application is the calculation of the moment of inertia $I_{d}$ of a uniform thin-walled spherical shell, of mass M and radius R, about a diameter. Taking the centre as the origin of coordinates, have $I_{x} = I_{y} = I_{z} = I_{d}$ , and $r_{i} = R$ . The theorem gives $3I_{d} = 2\Sigma_{i} m_{i}R^{2} = 2(\Sigma_{i}m_{i})R^{2} = 2MR^{2}$ , where $I_{d} = 2MR^{2/3}$ . Consider a solid cube of mass m and side L. What will be the value of $\sum m_{i}r_{i}^{2}$ for this body when the point of intersection of axes is the centre of the cube:

A

$\frac{ML^{2}}{2}$

B

$\frac{ML^{2}}{4}$

C

$\frac{ML^{2}}{3}$

D

$\frac{ML^{2}}{6}$

Question 140: Palette 140/197

Allen MCQ

Comprehension Passage: PARAGRAPH FOR QUESTION NO. 111 AND 112

In the treatment of moments of inertia, introductory textbooks often present two theorems, generally called the parallel axis theorem and the perpendicular axis theorem. There is another theorem of this same genre, which is not usually included, but which is interesting and useful. It is $$ I _ {x} + I _ {y} + I _ {z} = 2 \sum_ {i} m _ {i} r _ {i} ^ {2} $$ Here, $I_{x}$ , $I_{y}$ and $I_{z}$ are the moments of inertia about three mutually perpendicular intersecting axes, $m_{1}$ is the mass of the $i^{th}$ particle and $r_{i}$ is the distance from the intersection. The proof is simple: Taking the three axes as coordinate axes, have : $$ \begin{array}{r l} & I _ {x} + I _ {y} + I _ {z} \\ & = \sum_ {i} m _ {i} (y _ {i} ^ {2} + z _ {i} ^ {2}) + \sum_ {i} m _ {i} (z _ {i} ^ {2} + x _ {i} ^ {2}) + \sum_ {i} m _ {i} (x _ {i} ^ {2} + y _ {i} ^ {2}) \\ & = 2 \sum_ {i} m _ {i} (x _ {i} ^ {2} + y _ {i} ^ {2} + z _ {i} ^ {2}) = 2 \sum_ {i} m _ {i} r _ {i} ^ {2} \end{array} $$ One important application is the calculation of the moment of inertia $I_{d}$ of a uniform thin-walled spherical shell, of mass M and radius R, about a diameter. Taking the centre as the origin of coordinates, have $I_{x} = I_{y} = I_{z} = I_{d}$ , and $r_{i} = R$ . The theorem gives $3I_{d} = 2\Sigma_{i} m_{i}R^{2} = 2(\Sigma_{i}m_{i})R^{2} = 2MR^{2}$ , where $I_{d} = 2MR^{2/3}$ . Find the moment of inertia of ring of mass m and radius R about an axis passing through its centre and making an angle $45^{\circ}$ with its plane :

A

$\frac{MR^{2}}{4}$

B

$\frac{MR^{2}}{2}$

C

$\frac{3}{4} MR^2$

D

$MR^2$

Question 141: Palette 141/197

Allen MCQ

Comprehension Passage: PARAGRAPH FOR QUESTION NO. 113 TO 116

The figure shows an isosceles triangular plate of mass M and base L. The angle at the apex is $90^{\circ}$ . The apex lies at the origin and the base is parallel to X-axis The moment of inertia of the plate about the z-axis is:

A

$\frac{ML^{2}}{12}$

B

$\frac{ML^{2}}{24}$

C

$\frac{ML^{2}}{6}$

D

none of these

Question 142: Palette 142/197

Allen MCQ

Comprehension Passage: PARAGRAPH FOR QUESTION NO. 113 TO 116

The figure shows an isosceles triangular plate of mass M and base L. The angle at the apex is $90^{\circ}$ . The apex lies at the origin and the base is parallel to X-axis The moment of inertia of the plate about the x-axis is :-

A

$\frac{ML^{2}}{8}$

B

$\frac{ML^{2}}{32}$

C

$\frac{ML^{2}}{24}$

D

$\frac{ML^{2}}{6}$

Question 143: Palette 143/197

Allen MCQ

Comprehension Passage: PARAGRAPH FOR QUESTION NO. 113 TO 116

The figure shows an isosceles triangular plate of mass M and base L. The angle at the apex is $90^{\circ}$ . The apex lies at the origin and the base is parallel to X-axis The moment of inertia of the plate about its base parallel to the x-axis is :-

A

$\frac{ML^2}{18}$

B

$\frac{ML^2}{36}$

C

$\frac{ML^2}{24}$

D

none of these

Question 144: Palette 144/197

Allen MCQ

Comprehension Passage: PARAGRAPH FOR QUESTION NO. 113 TO 116

The figure shows an isosceles triangular plate of mass M and base L. The angle at the apex is $90^{\circ}$ . The apex lies at the origin and the base is parallel to X-axis The moment of inertia of the plate about the y-axis is:- $\tau = I\alpha$ Torque

A

$\frac{ML^2}{6}$

B

$\frac{ML^2}{8}$

C

$\frac{ML^2}{24}$

D

None of these

Question 145: Palette 145/197

Allen MCQ

Comprehension Passage: PARAGRAPH FOR QUESTION 117 TO 119

A simplified model of a bicycle of mass M has two tyres that each comes into contact with the ground at a point. The wheelbase of this bicycle (the distance between the points of contact with the ground) is w, and the center of mass of the bicycle is located midway between the tyres and at height h above the ground. Air resistance may be ignored. Case - 1 (Q. 117): Assume that cycle is moving rightwards & has acceleration a. Both tyres roll purely. Case - 2 (Q. 118 and 119): The bicycle is moving to the right, but slowing down at a constant rate. The acceleration has a magnitude a. Assume that the coefficient of sliding friction between each tyre and the ground is $\mu$ , and that both tyres are skidding (sliding without rotating). Express your answers in terms of w, h, M and g. The direction of friction on the tyres is

A

forward on both

B

backward on both

C

forward on front & backward on rear

D

forward on rear & backward on front.

Question 146: Palette 146/197

Allen MCQ

Comprehension Passage: PARAGRAPH FOR QUESTION 117 TO 119

A simplified model of a bicycle of mass M has two tyres that each comes into contact with the ground at a point. The wheelbase of this bicycle (the distance between the points of contact with the ground) is w, and the center of mass of the bicycle is located midway between the tyres and at height h above the ground. Air resistance may be ignored. Case - 1 (Q. 117): Assume that cycle is moving rightwards & has acceleration a. Both tyres roll purely. Case - 2 (Q. 118 and 119): The bicycle is moving to the right, but slowing down at a constant rate. The acceleration has a magnitude a. Assume that the coefficient of sliding friction between each tyre and the ground is $\mu$ , and that both tyres are skidding (sliding without rotating). Express your answers in terms of w, h, M and g. What is the maximum value of $\mu$ so that both tyres remain in contact with the ground?

A

$\frac{w}{2h}$

B

$\frac{h}{2w}$

C

$\frac{2h}{w}$

D

$\frac{w}{h}$

Question 147: Palette 147/197

Allen MCQ

Comprehension Passage: PARAGRAPH FOR QUESTION 117 TO 119

A simplified model of a bicycle of mass M has two tyres that each comes into contact with the ground at a point. The wheelbase of this bicycle (the distance between the points of contact with the ground) is w, and the center of mass of the bicycle is located midway between the tyres and at height h above the ground. Air resistance may be ignored. Case - 1 (Q. 117): Assume that cycle is moving rightwards & has acceleration a. Both tyres roll purely. Case - 2 (Q. 118 and 119): The bicycle is moving to the right, but slowing down at a constant rate. The acceleration has a magnitude a. Assume that the coefficient of sliding friction between each tyre and the ground is $\mu$ , and that both tyres are skidding (sliding without rotating). Express your answers in terms of w, h, M and g. What is the maximum value of a so that both tyres remain in contact with the ground?

A

$\frac{wg}{h}$

B

$\frac{wg}{2h}$

C

$\frac{hg}{2w}$

D

$\frac{2hg}{w}$

Question 148: Palette 148/197

Allen MCQ

Comprehension Passage: PARAGRAPH FOR QUESTION NO. 120 AND 121

A thread is wrapped around a cylindrical spool of radius 1.5 cm whose central axis is fixed on a support. The thread is pulled with a constant acceleration. It is seen that it takes 2 sec to pull 6.28 cm of thread starting from rest. What is the angular velocity of the spool at 1.5 sec?

A

$\pi$ rad/s

B

$\frac{\pi}{2}$ rad/s

C

$\frac{3\pi}{4}$ rad/s

D

$\frac{\pi}{4}$ rad/s

Question 149: Palette 149/197

Allen MCQ

Comprehension Passage: PARAGRAPH FOR QUESTION NO. 120 AND 121

A thread is wrapped around a cylindrical spool of radius 1.5 cm whose central axis is fixed on a support. The thread is pulled with a constant acceleration. It is seen that it takes 2 sec to pull 6.28 cm of thread starting from rest. If the thread was pulled with a constant speed so that it takes 2 sec to pull 6.28 cm of thread. The angular velocity of spool at 1.5 sec is:

A

$\frac{4}{3}$ of what was obtained in previous question.

B

$\frac{2}{3}$ of what was obtained in previous question.

C

Same as what was obtained in previous question.

D

Cannot be predicted

Question 150: Palette 150/197

Allen MCQ

Comprehension Passage: PARAGRAPH FOR QUESTION NO. 122 AND 123

When an athlete takes off from the ground in a long jump, the force on the launching foot gives her an angular momentum in clockwise direction around the horizontal axis. This rotation would not allow her to land properly. In the landing, the legs should be together and extended forward at an angle so that heels marks the sand at greatest distance. Once in air the angular momentum cannot change. However, the jumper can shift most of the angular momentum to the arms by rotating them clockwise as shown in the figure. Then the body comes to proper orientation for landing. Assume that in 0.7 sec one arm covers 1/2 a revolution and another arm covers 1 revolution. Each arm is a thin rod of mass 4 kg and length 0.6 m rotating at one end. Assume that this 0.7 sec. is time of flight. What is the angular momentum of both arms in athlete's frame (in SI units)

A

$\frac{14.4\pi}{7}$

B

$\frac{7.2\pi}{7}$

C

$\frac{4.8\pi}{7}$

D

$\frac{2.4\pi}{7}$

Question 151: Palette 151/197

Allen MCQ

Comprehension Passage: PARAGRAPH FOR QUESTION NO. 122 AND 123

When an athlete takes off from the ground in a long jump, the force on the launching foot gives her an angular momentum in clockwise direction around the horizontal axis. This rotation would not allow her to land properly. In the landing, the legs should be together and extended forward at an angle so that heels marks the sand at greatest distance. Once in air the angular momentum cannot change. However, the jumper can shift most of the angular momentum to the arms by rotating them clockwise as shown in the figure. Then the body comes to proper orientation for landing. Assume that in 0.7 sec one arm covers 1/2 a revolution and another arm covers 1 revolution. Each arm is a thin rod of mass 4 kg and length 0.6 m rotating at one end. Assume that this 0.7 sec. is time of flight. If rest of the athlete's body is assumed to be a rigid body having a moment of inertia of $18 \, kg m^{2}$ about its centre of mass, what is its change in angular velocity due to this rotation of arms?

A

$\frac{\pi}{35} \, rad/s$

B

$\frac{2\pi}{35} \, rad/s$

C

$\frac{8\pi}{35} \, rad/s$

D

$\frac{4\pi}{35} \, rad/s$
Dynamics

Question 152: Palette 152/197

Allen MCQ

Comprehension Passage: PARAGRAPH FOR QUESTION NO. 124 TO 126

In the following problems, indicate the correct direction of friction force acting on the cylinder, which is pulled on a rough surface by a constant force F. A cylinder of mass M and radius R is pulled horizontally by a force F. The friction force can be given by which of the following diagrams :-

A

B

C

$$<br /> f = 0<br /> $$

D

Cannot be interpreted

Question 153: Palette 153/197

Allen MCQ

Comprehension Passage: PARAGRAPH FOR QUESTION NO. 124 TO 126

In the following problems, indicate the correct direction of friction force acting on the cylinder, which is pulled on a rough surface by a constant force F. A cylinder is pulled horizontally by a force F acting at a point below the centre of mass of the cylinder, as shown in figure. The friction force can be given by which of the following diagrams :-

A

B

C

D

cannot be interpreted

Question 154: Palette 154/197

Allen MCQ

Comprehension Passage: PARAGRAPH FOR QUESTION NO. 124 TO 126

In the following problems, indicate the correct direction of friction force acting on the cylinder, which is pulled on a rough surface by a constant force F. A cylinder is pulled horizontally by a force F acting at a point above the centre of mass of the cylinder, as shown in figure. The friction force can be given by which of the following diagrams

A

B

C

D

cannot be interpreted

Question 155: Palette 155/197

Allen MCQ

Comprehension Passage: PARAGRAPH FOR QUESTION NO. 127 AND 128

A disc of mass m and radius R is placed over a plank of same mass m. There is sufficient friction between disc and plank to prevent slipping. A force F is applied at the centre of the disc. Acceleration of the plank is :-

A

$\frac{F}{2m}$

B

$\frac{3F}{4m}$

C

$\frac{F}{4m}$

D

$\frac{3F}{2m}$

Question 156: Palette 156/197

Allen MCQ

Comprehension Passage: PARAGRAPH FOR QUESTION NO. 127 AND 128

A disc of mass m and radius R is placed over a plank of same mass m. There is sufficient friction between disc and plank to prevent slipping. A force F is applied at the centre of the disc. Force of friction between the disc and the plank is :-

A

$\frac{F}{2}$

B

$\frac{F}{4}$

C

$\frac{F}{3}$

D

$\frac{2F}{3}$

Question 157: Palette 157/197

Allen MCQ

Comprehension Passage: PARAGRAPH FOR QUESTION NO. 129 TO 131

A uniform rod of mass 4m and length L lies on a smooth horizontal table. A particle of mass m moving on the table with speed $v_{0}$ strikes the rod perpendicularly at an end and stops. Velocity of centre of mass of the rod after collision is

A

$v_{0}$

B

$\frac{v_{0}}{2}$

C

$\frac{v_{0}}{4}$

D

$\frac{v_{0}}{6}$

Question 158: Palette 158/197

Allen MCQ

Comprehension Passage: PARAGRAPH FOR QUESTION NO. 129 TO 131

A uniform rod of mass 4m and length L lies on a smooth horizontal table. A particle of mass m moving on the table with speed $v_{0}$ strikes the rod perpendicularly at an end and stops. Angular velocity of the rod after collision is

A

$\frac{3v_0}{2L}$

B

$\frac{3v_0}{4L}$

C

$\frac{v_0}{2L}$

D

$\frac{v_0}{4L}$

Question 159: Palette 159/197

Allen MCQ

Comprehension Passage: PARAGRAPH FOR QUESTION NO. 129 TO 131

A uniform rod of mass 4m and length L lies on a smooth horizontal table. A particle of mass m moving on the table with speed $v_{0}$ strikes the rod perpendicularly at an end and stops. Distance travelled by the centre of rod by the time it turns through one revolution is

A

$\pi L$

B

$2\pi L$

C

$\frac{\pi L}{12}$

D

$\frac{\pi L}{3}$

Question 160: Palette 160/197

Allen MSQ

Comprehension Passage: PARAGRAPH FOR QUESTION NO. 132 TO 135

A ring of mass M and radius R sliding with a velocity $v_{0}$ suddenly enters into rough surface where the coefficient of friction is $\mu$ , as shown in figure. Choose the correct statement(s)

A

As the ring enters on the rough surface, the limiting friction force acts on it.

B

The direction of friction is opposite to the direction of motion.

C

The friction force accelerates the ring in the clockwise sense about its centre of mass.

D

As the ring enters on the rough surface it starts rolling.

Question 161: Palette 161/197

Allen MCQ

Comprehension Passage: PARAGRAPH FOR QUESTION NO. 132 TO 135

A ring of mass M and radius R sliding with a velocity $v_{0}$ suddenly enters into rough surface where the coefficient of friction is $\mu$ , as shown in figure. Choose the correct statement(s)

A

The momentum of the ring is conserved

B

The angular momentum of the ring is conserved about its centre of mass

C

The angular momentum of the ring conserved about any point on the horizontal surface in line of friction.

D

The mechanical energy of the ring is conserved

Question 162: Palette 162/197

Allen MSQ

Comprehension Passage: PARAGRAPH FOR QUESTION NO. 132 TO 135

A ring of mass M and radius R sliding with a velocity $v_{0}$ suddenly enters into rough surface where the coefficient of friction is $\mu$ , as shown in figure. Choose the correct statement(s):

A

The ring starts its rolling motion when the centre of mas stationary.

B

The ring starts rolling motion when the point of contact becomes stationary.

C

The time after which the ring starts rolling is $\frac{v_{0}}{2\mu g}$ .

D

The rolling velocity is $\frac{v_{0}}{2}$ .

Question 163: Palette 163/197

Allen MSQ

Comprehension Passage: PARAGRAPH FOR QUESTION NO. 132 TO 135

A ring of mass M and radius R sliding with a velocity $v_{0}$ suddenly enters into rough surface where the coefficient of friction is $\mu$ , as shown in figure. Choose the correct alternative(s) :-

A

The linear distance moved by the centre of mass before the ring starts rolling is $\frac{3v_{0}^{2}}{8\mu g}$ .

B

The net work done by friction force is $-\frac{3}{8} mv_0^2$ .

C

The loss is kinetic energy of the ring is $\frac{mv_0^2}{4}$ .

D

The gain in rotational kinetic energy is $+\frac{mv_0^2}{8}$ .

Question 164: Palette 164/197

Allen MCQ

A particle starts from the point $(0m, 8m)$ and moves with uniform velocity of $3\hat{i}m/s$ . After 5 seconds, the angular velocity of the particle about the origin will be:

A

$\frac{8}{289}\mathrm{rad} / s$

B

$\frac{3}{8}\mathrm{rad} / s$

C

$\frac{24}{289}\mathrm{rad} / s$

D

$\frac{8}{17}\mathrm{rad} / s$

Question 165: Palette 165/197

Allen MCQ

Two points of a rigid body are moving as shown. The angular velocity of the body is:

A

$\frac{v}{2R}$

B

$\frac{v}{R}$

C

$\frac{2v}{R}$

D

$\frac{2v}{3R}$

Question 166: Palette 166/197

Allen MCQ

A non-uniform sphere can be kept on a rough inclined plane so that it is in equilibrium. In the figure below the dots represents location of centre of mass. In which one of the positions can sphere be in equilibrium?

A

A

B

B

C

C

D

D

Question 167: Palette 167/197

Allen MCQ

A uniform flag pole of length L and mass M is pivoted on the ground with a frictionless hinge. The flag pole makes an angle $\theta$ with the horizontal. The moment of inertia of the flag pole about one end is $(1/3)ML^{2}$ . If it starts falling from the position shown in the accompanying figure, the linear acceleration of the free end of the flag pole (labeled P) immediately after it starts falling off would be:

A

$(2/3)g\cos\theta$

B

$(2/3)g$

C

g

D

$(3/2)g\cos\theta$

Question 168: Palette 168/197

Allen MCQ

A solid sphere with a velocity (of centre of mass) v and angular velocity $\omega$ is gently placed on a rough horizontal surface. The frictional force on the sphere:

A

must be forward (in direction of v)

B

must be backward (opposite to v)

C

cannot be zero

D

none of the above

Question 169: Palette 169/197

Allen MCQ

A hoop and a solid cylinder have the same mass and radius. They both roll, without slipping, on a horizontal surface. If their kinetic energies are equal

A

the hoop has a greater translational speed than the cylinder

B

the cylinder has a greater translational speed than the hoop

C

the hoop and the cylinder have the same translational speed

D

the hoop has a greater rotational speed than the cylinder

Question 170: Palette 170/197

Allen MCQ

The moment of inertia of a solid cylinder about its axis is given by $(1/2)MR^{2}$ . If this cylinder rolls without slipping, the ratio of its rotational kinetic energy to its translational kinetic energy is

A

1 : 1

B

2 : 2

C

1 : 2

D

1 : 3

Question 171: Palette 171/197

Allen MCQ

A rod of weight w is supported by two parallel knives at end points A and B and is in equilibrium in a horizontal position. The knives are at a distance 'd' from each other. The centre of mass of the rod is at a distance 'x' from A.

A

the normal reaction at A is $\frac{wx}{d}$

B

the normal reaction at A is $\frac{w(d-x)}{2d}$

C

the normal reaction at B is $\frac{wx}{d}$

D

the normal reaction at B is $\frac{w(d-x)}{d}$

Question 172: Palette 172/197

Allen MCQ

A man spinning in free space changes the shape of his body, eg. by spreading his arms or curling up. By doing this, he can not change his

A

moment of inertia

B

angular momentum

C

angular velocity

D

rotational kinetic energy

Question 173: Palette 173/197

Allen MCQ

A heavy seesaw (i.e., not massless) is out of balance. A light girl sits on the end that is tilted downward, and a heavy body sits on the other side so that the seesaw now balances. If they both move forward so that they are one-half their original distance from the pivot point (the fulcrum) what will happen to the seesaw?

A

The side the body is sitting on will tilt downward

B

The side the girl is sitting on will once again tilt downward

C

Nothing; the seesaw will still be balanced

D

It is impossible to say without knowing the masses and the distances

Question 174: Palette 174/197

Allen MCQ

A rod is hinged at its centre and rotated by applying a constant torque starting from rest. The power developed by the external torque as a function of time is:

A

B

C

D

Question 175: Palette 175/197

Allen MCQ

A hollow smooth uniform sphere A of mass ‘m’ rolls without sliding on a smooth horizontal surface. It collides head on elastically with another stationary smooth solid sphere B of the same mass m and same radius. The ratio of kinetic energy of ‘B’ to that of ‘A’ just after the collision is :

A

1:1

B

2:3

C

3:2

D

None

Question 176: Palette 176/197

Allen MCQ

A wheel of radius r rolling on a straight line, the velocity of its centre being v. At a certain instant the point of contact of the wheel with the grounds is M and N is the highest point on the wheel (diametrically opposite to M). The incorrect statement is:

A

The velocity of any point P of the wheel is proportional to MP.

B

Points of the wheel moving with velocity greater than v form a larger area of the wheel than points moving with velocity less than v.

C

The point of contact $M$ is instantaneously at rest.

D

The velocities of any two parts of the wheel which are equidistant from centre are equal.

Question 177: Palette 177/197

Allen MCQ

Particles of masses 1, 2 and 3kg respectively are arranged at the corners of an equilateral triangle as shown in figure. Obtain the moment of inertia of the system about an axis passing through the centre of mass of the system and perpendicular to its plane. The side of the triangle is 1m.

A

$4kg - m^{2}$

B

$\frac{5}{6} kg\cdot m^{2}$

C

$\frac{11}{6} kg - m^2$

C

5 kg-m²

Question 178: Palette 178/197

Allen MCQ

Inner and outer radii of a spool are r and R respectively. A thread is wound over its inner surface and placed over a rough horizontal surface. Thread is pulled by a force F as shown in fig. then in case of pure rolling:

A

Thread unwinds, spool rotates anticlockwise and friction act leftwards.

B

Thread winds, spool rotates clockwise and friction acts leftwards.

C

Thread winds, spool moves to the right and friction act rightwards.

D

Thread winds, spool moves to the right and friction does not come into existence.

Question 179: Palette 179/197

Allen MCQ

A yo-yo is resting on a perfectly rough horizontal table. Forces $F_{1}, F_{2}$ and $F_{3}$ are applied separately as shown. The correct statement is :-

A

when $F_{3}$ is applied the centre of mass will move to the right.

B

when $F_{2}$ is applied the centre of mass will move to the left.

C

when $F_{1}$ is applied the centre of mass will move to the right.

D

when $F_{2}$ is applied the centre of mass will move to the right.

Question 180: Palette 180/197

Allen MCQ

Pulley is rotating with angular velocity $\omega$ . Pulley is sufficiently rough so that string does not slip on the pulley. Then angular momentum of system (pulley + blocks) about axle of pulley is :-

A

$I\omega$

B

$(m_{1} + m_{2} + I)\omega r$

C

$(m_{1} + m_{2})r^{2}\omega +I\omega$

D

$(m_{1}r^{2} + m_{2}r^{2} + I)\omega^{2}$

Question 181: Palette 181/197

Allen MCQ

The free end of the string wound on the surface of a solid cylinder of mass M and radius r is pulled up by a force F as shown. If there is sufficient friction between cylinder and floor so that the cylinder is able to roll without slipping, find the maximum angular acceleration that the cylinder can have (without cylinder leaving contact with surface).

A

$\frac{g}{r}$

B

$\frac{2g}{3r}$

C

$\frac{4g}{3r}$

Question 182: Palette 182/197

Allen MCQ

From a solid sphere of mass M and radius R, a cube of maximum possible volume is cut. Moment of inertia of cube about an axis passing through its centre and perpendicular to one of its faces is:-

A

$\frac{4MR^{2}}{9\sqrt{3}\pi}$

B

$\frac{4MR^{2}}{3\sqrt{3}\pi}$

C

$\frac{MR^{2}}{32\sqrt{2}\pi}$

D

$\frac{MR^{2}}{16\sqrt{2}\pi}$

Question 183: Palette 183/197

Allen MCQ

A roller is made by joining together two cones at their vertices O. It is kept on two rails AB and CD which are placed asymmetrically (see figure), with its axis perpendicular to CD and its centre O at the centre of line joining AB and CD (see figure). It is given a light push so that it starts rolling with its centre O moving parallel to CD in the direction shown. As it moves, the roller will tend to :-

A

turn left and right alternately

B

turn left

C

turn right

D

go straight

Question 184: Palette 184/197

Allen MCQ

A small sphere $A$ of mass $m$ and radius $r$ rolls without slipping inside a large fixed hemispherical bowl of radius $R (>> r)$ as shown in figure. If the sphere starts from rest at the top point of the hemisphere find the normal force exerted by the small sphere on the hemisphere when it is at the bottom $B$ of the hemisphere. $$ \frac {7}{5} m g $$

A

$\frac{10}{7} mg$

B

$\frac{17}{7} mg$

C

$\frac{5}{7} mg$

D

$\frac{7}{5} m$

Question 185: Palette 185/197

Allen MCQ

A straight rod of length L is released on a frictionless horizontal floor in a vertical position. As it falls + slips, the distance of a point on the rod from the lower end, which follows a quarter circular locus is :-

A

L/2

B

L/4

C

L/8

D

None

Question 186: Palette 186/197

Allen MSQ

A ring rolls without slipping on the ground. Its centre C moves with a constant speed u. P is any point on the ring. The speed of P with respect to the ground is v.

A

$0 \leq n \leq 2u$

B

v = u, if CP is horizontal

C

v = u, if CP makes an angle of $30^{\circ}$ with the horizontal and P is below the horizontal level of C.

D

$v = \sqrt{2} u$ , if $CP$ is horizontal

Question 187: Palette 187/197

Allen MSQ

A disc of circumference s is at rest at a point A on a horizontal surface when a constant horizontal force begins to act on its centre. Between A and B there is sufficient friction to prevent slipping, and the surface is smooth to the right of B. AB = s. The disc moves from A to B in time T. To the right of B,

A

the angular acceleration of the disc will disappear, linear acceleration will remain unchanged.

B

linear acceleration of the disc will increase.

C

the disc will make one rotation in time T/2.

D

the disc will cover a distance greater than s in further time T.

Question 188: Palette 188/197

Allen MSQ

A body is in equilibrium under the influence of a number of forces. Each force has a different line of action. The minimum number of forces required is

A

2, if their lines of action pass through the centre of mass of the body.

B

3, if their lines of action are not parallel.

C

3, if their lines of action are parallel.

D

4, if their lines of action are parallel and all the forces have the same magnitude.

Question 189: Palette 189/197

Allen MCQ

Comprehension Passage: PARAGRAPH FOR QUESTION NO. 6 TO 8

In figure, the winch is mounted on an axle, and the 6-sided nut is welded to the winch. By turning the nut with a wrench, a person can rotate the winch. For instance, turning the nut clockwise lifts the block off the ground, because more and more rope gets wrapped around the winch. Three students agree that using a longer wrench makes it easier to turn the winch. But they disagree about why. All three students are talking about the case where the winch is used, over a 10 s time interval, to lift the block one metre off the ground. Student 1: By using a longer wrench, the person decreases the average force he must exert on the wrench, in order to lift the block one metre in 10s. Student 2: Using a longer wrench reduces the work done by the person as he uses the winch to lift the block 1m in 10s. Student 3 : Using a longer wrench reduces the power that the person must exert to lift the block 1m in 10s. Student 1 is :-

A

correct, because the torque that the wrench must exert to lift the block doesn't depend on the wrench's length.

B

correct, because using a longer wrench decreases the torque it must exert on the winch.

C

incorrect, because the torque that the wrench must exert to lift the block doesn't depend on the wrench's length.

D

Incorrect, because using a longer wrench decreases the torque it must exert on the winch.

Question 190: Palette 190/197

Allen MCQ

Comprehension Passage: PARAGRAPH FOR QUESTION NO. 6 TO 8

In figure, the winch is mounted on an axle, and the 6-sided nut is welded to the winch. By turning the nut with a wrench, a person can rotate the winch. For instance, turning the nut clockwise lifts the block off the ground, because more and more rope gets wrapped around the winch. Three students agree that using a longer wrench makes it easier to turn the winch. But they disagree about why. All three students are talking about the case where the winch is used, over a 10 s time interval, to lift the block one metre off the ground. Student 1: By using a longer wrench, the person decreases the average force he must exert on the wrench, in order to lift the block one metre in 10s. Student 2: Using a longer wrench reduces the work done by the person as he uses the winch to lift the block 1m in 10s. Student 3 : Using a longer wrench reduces the power that the person must exert to lift the block 1m in 10s. Which of the following is true about student 2 and 3 :

A

Student 2 and 3 are both correct

B

Student 2 is correct, but student 3 is incorrect

C

Student 3 is correct, but student 2 is incorrect

D

Student 2 and 3 are both incorrect

Question 191: Palette 191/197

Allen MCQ

Comprehension Passage: PARAGRAPH FOR QUESTION NO. 6 TO 8

In figure, the winch is mounted on an axle, and the 6-sided nut is welded to the winch. By turning the nut with a wrench, a person can rotate the winch. For instance, turning the nut clockwise lifts the block off the ground, because more and more rope gets wrapped around the winch. Three students agree that using a longer wrench makes it easier to turn the winch. But they disagree about why. All three students are talking about the case where the winch is used, over a 10 s time interval, to lift the block one metre off the ground. Student 1: By using a longer wrench, the person decreases the average force he must exert on the wrench, in order to lift the block one metre in 10s. Student 2: Using a longer wrench reduces the work done by the person as he uses the winch to lift the block 1m in 10s. Student 3 : Using a longer wrench reduces the power that the person must exert to lift the block 1m in 10s. If several wrenches all apply the same torque to a nut, which graph best expresses the relationship between the force the person must apply to the wrench, and the length of the wrench :-

A

B

C

D

Question 192: Palette 192/197

Allen MCQ

Comprehension Passage: PARAGRAPH FOR QUESTION NO. 9 AND 10

A uniform rod is fixed to a rotating turntable so that its lower end is on the axis of the turntable and it makes an angle of $20^{\circ}$ to the vertical. (The rod is thus rotating with uniform angular velocity about a vertical axis passing through one end.) If the turntable is rotating clockwise as seen from above. What is the direction of the rod's angular momentum vector (calculated about its lower end)?

A

vertically downwards

B

down at $20^{\circ}$ to the horizontal

C

up at $20^{\circ}$ to the horizontal

D

vertically upwards

Question 193: Palette 193/197

Allen MCQ

Comprehension Passage: PARAGRAPH FOR QUESTION NO. 9 AND 10

A uniform rod is fixed to a rotating turntable so that its lower end is on the axis of the turntable and it makes an angle of $20^{\circ}$ to the vertical. (The rod is thus rotating with uniform angular velocity about a vertical axis passing through one end.) If the turntable is rotating clockwise as seen from above. Is there a torque acting on it, and if so in what direction?

A

yes, vertically

B

yes, horizontally

C

yes at $20^{\circ}$ to the horizontal

D

no

Question 194: Palette 194/197

Allen MCQ

Comprehension Passage: PARAGRAPH FOR QUESTION NO. 11 TO 14

A spring having initial unstretched length $\ell_{0}$ is lying on a smooth table. It's one end is fixed and the other one is fastened to a small particle of mass m. The particle is imparted an initial speed $v_{0}$ horizontally in a direction perpendicular to the spring. In the course of the motion in horizontal plane, the maximum elongation of the spring is $\Delta\ell = \ell_{0}/10$ . (Given: m = 0.1 kg, $\ell_{0} = 1m$ , $v_{0} = 11m/s$ ). In the course of motion, which of the quantities relating to spring block system are conserved?

A

kinetic energy

B

momentum

C

angular momentum

D

potential energy

Question 195: Palette 195/197

Allen MCQ

Comprehension Passage: PARAGRAPH FOR QUESTION NO. 11 TO 14

A spring having initial unstretched length $\ell_{0}$ is lying on a smooth table. It's one end is fixed and the other one is fastened to a small particle of mass m. The particle is imparted an initial speed $v_{0}$ horizontally in a direction perpendicular to the spring. In the course of the motion in horizontal plane, the maximum elongation of the spring is $\Delta\ell = \ell_{0}/10$ . (Given: m = 0.1 kg, $\ell_{0} = 1m$ , $v_{0} = 11m/s$ ). Which of the following is correct about initial situation and situation at maximum elongation?

A

the orientation of spring in both positions should be perpendicular to each other.

B

the velocity at maximum extension should be zero.

C

the velocity at maximum extension as well as at initial position should be perpendicular to spring.

D

Acceleration should be zero at the maximum extension as well as at initial position.

Question 196: Palette 196/197

Allen MCQ

Comprehension Passage: PARAGRAPH FOR QUESTION NO. 11 TO 14

A spring having initial unstretched length $\ell_{0}$ is lying on a smooth table. It's one end is fixed and the other one is fastened to a small particle of mass m. The particle is imparted an initial speed $v_{0}$ horizontally in a direction perpendicular to the spring. In the course of the motion in horizontal plane, the maximum elongation of the spring is $\Delta\ell = \ell_{0}/10$ . (Given: m = 0.1 kg, $\ell_{0} = 1m$ , $v_{0} = 11m/s$ ). Student-A: At maximum extension $mv'(\ell_{0} + \Delta\ell) = mv_{0}\ell_{0}$ . Student-B: at maximum extension $k\Delta\ell = \frac{mv'^{2}}{\left(\ell_{0} + \Delta\ell\right)}$ . Student-C: $\frac{1}{2} mv_0^2 = \frac{1}{2} mv'^2 + \frac{1}{2} k\Delta \ell^2$ . Where v' is velocity at instant of maximum extension:

A

Only Student-A and B are correct

B

Only Student-A and C are correct

C

Only Student-B and C are correct

D

All are correct

Question 197: Palette 197/197

Allen MCQ

Comprehension Passage: PARAGRAPH FOR QUESTION NO. 11 TO 14

A spring having initial unstretched length $\ell_{0}$ is lying on a smooth table. It's one end is fixed and the other one is fastened to a small particle of mass m. The particle is imparted an initial speed $v_{0}$ horizontally in a direction perpendicular to the spring. In the course of the motion in horizontal plane, the maximum elongation of the spring is $\Delta\ell = \ell_{0}/10$ . (Given: m = 0.1 kg, $\ell_{0} = 1m$ , $v_{0} = 11m/s$ ). The value of spring constant (in N/m) is:

A

100

B

210

C

420

D

105