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Rotational Motion

166 Questions MCQ (Single Correct) Start Allen Test
All Exams Allen Advance Errorless
JEE (Advanced) 2023 Q1 Allen JEE Advanced PYQ 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 $^{-1}$
JEE (Advanced) 2022 Q2 Allen JEE Advanced PYQ 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.
$\frac{9}{2}$
JEE (Main) 2021 Q3 Allen JEE Main PYQ 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.
$-15 + 20\sqrt{3}, 15 + 20\sqrt{3}$
JEE (Main) 2021 Q4 Allen Rotational Motion 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
JEE (Main) 2021 Q5 Allen Rotational Motion 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.
$\frac{\left(\omega_{1}-\omega_{2}\right)^{2}}{2\left(I_{1}+I_{2}\right)}$
JEE (Main) 2021 Q6 Allen Rotational Motion 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.
$1.49 \times 10^{2} \, kg / m^{3}$
JEE (Main) 2020 Q7 Allen JEE Main PYQ 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.
$Mg\sqrt{1-\left(\frac{R-a}{R}\right)^{2}}$
JEE (Main) 2020 Q8 Allen JEE Main PYQ 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.
$4 9 ^ {\circ}$
JEE (Main) 2020 Q9 Allen Rotational Motion 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.
$2\pi a^{4}\left(\frac{A}{4}+\frac{B}{5}\right)$
JEE (Advanced) 2020 Q10 Allen JEE Advanced PYQ 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) ā€“
A.
$\sin \theta = \frac{r}{R}$
B.
$\tan \theta = \frac{r}{R}$
C.
$\sin \theta = \frac{r}{2R}$
D.
$\cos\theta=\frac{r}{2R}$
JEE (Advanced) 2020 Q11 Allen JEE Advanced PYQ 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.

JEE (Main) 2019 Q12 Allen JEE Main PYQ 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.
$\mu FR/6$
JEE (Main) 2019 Q13 Allen JEE Main PYQ 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.
$I = \frac{15}{16} I_0$
JEE (Main) 2019 Q14 Allen JEE Main PYQ 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
JEE (Main) 2019 Q15 Allen JEE Main PYQ 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.
$\frac{2k}{I}\theta$
JEE (Main) 2017 Q16 Allen JEE Main PYQ 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.
$\frac{\sqrt{3}}{2}$
JEE (Advanced) 2017 Q17 Allen JEE Advanced PYQ 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.
JEE (Main) 2016 Q18 Allen JEE Main PYQ 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 $C$
JEE (Advanced) 2016 Q19 Allen JEE Advanced PYQ 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.
$\frac{R}{4}(e^{\omega t} + e^{-\omega t})$
JEE (Advanced) 2016 Q20 Allen JEE Advanced PYQ 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.
$\frac{1}{2}m\omega^{2}R\left(e^{2\omega t} - e^{-2\omega t}\right)\hat{j} + mg\hat{k}$
JEE (Advanced) 2015 Q21 Allen JEE Advanced PYQ MCQ
The densities of two solid spheres A and B of the same radii R vary with radial distance r as $\rho_{A}(r)=k\left(\frac{r}{R}\right)$ and $\rho_{B}(r)=k\left(\frac{r}{R}\right)^{5}$ , respectively, where k is a constant. The moments of inertia of the individual spheres about axes passing through their centres are $I_{A}$ and $I_{B}$ , respectively. If $\frac{I_{B}}{I_{A}}=\frac{n}{10}$ , the value of n is.
JEE (Main) 2014 Q22 Allen JEE Main PYQ MCQ
A mass ā€˜mā€™ 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.
$\frac{g}{2}$
JEE (Advanced) 2014 Q23 Allen JEE Advanced PYQ MCQ
A uniform circular disc of mass 1.5 kg and radius 0.5 m is initially at rest on a horizontal frictionless surface. Three forces of equal magnitude F = 0.5 N are applied simultaneously along the three sides of an equilateral triangle XYZ with its vertices on the perimeter of the disc (see figure). One second after applying the forces, the angular speed of the disc in rad s $^{-1}$ is
Q24 Allen Concept of $w, \alpha, v, a$ 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
Q25 Allen Concept of $w, \alpha, v, a$ 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