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Rotational Motion

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JEE (Main) 2020 Q26 Allen 15. JEE Main PYQ Numerical
ABC is a plane lamina of the shape of an equilateral triangle. D, E are mid points of AB, AC and G is the centroid of the lamina. Moment of inertia of the lamina about an axis passing through G and perpendicular to the plane ABC is $I_{0}$ . If part ADE is removed, the moment of inertia of the remaining part about the same axis is $\frac{NI_{0}}{16}$ where N is an integer. Value of N is ____.
JEE (Main) 2020 Q27 Allen 16. 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 (Main) 2020 Q28 Allen 16. Rotational Motion Numerical
Consider a uniform cubical box of side $a$ on a rough floor that is to be moved by applying minimum possible force $F$ at a point $b$ above its centre of mass (see figure). If the coefficient of friction is $\mu = 0.4$ , the maximum possible value of $100 \times \frac{b}{a}$ for a box not to topple before moving is ____.
JEE (Advanced) 2020 Q29 Allen 18. 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 Q30 Allen 18. 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 (Advanced) 2020 Q31 Allen 18. JEE Advanced PYQ Numerical
Put a uniform meter scale horizontally on your extended index fingers with the left one at 0.00 cm and the right one at 90.00 cm. When you attempt to move both the fingers slowly towards the center, initially only the left finger slips with respect to the scale and the right finger does not. After some distance, the left finger stops and the right one starts slipping. Then the right finger stops at a distance $x_{R}$ from the center (50.00 cm) of the scale and the left one starts slipping again. This happens because of the difference in the frictional forces on the two fingers. If the coefficients of static and dynamic friction between the fingers and the scale are 0.40 and 0.32, respectively, the value of $x_{R}$ (in cm) is ____.
JEE (Main) 2019 Q32 Allen 15. 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 Q33 Allen 15. 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 Q34 Allen 15. 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 Q35 Allen 15. 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 (Advanced) 2018 Q36 Allen 18. JEE Advanced PYQ 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.
$L = \sqrt{\frac{mk}{2}} R^{2}$
JEE (Main) 2017 Q37 Allen 15. 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 Q38 Allen 18. JEE Advanced PYQ 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.
JEE (Advanced) 2017 Q39 Allen 18. 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 Q40 Allen 15. 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 Q41 Allen 18. JEE Advanced PYQ 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 $\vec{\tau} = -\left(\frac{20}{3}\right)\hat{k}$ $Nm$ .
JEE (Advanced) 2016 Q42 Allen 18. 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 Q43 Allen 18. 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 Q44 Allen 18. 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 Q45 Allen 15. 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 Q46 Allen 18. 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
Q47 Allen 1. 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
Q48 Allen 1. 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
Q49 Allen 1. Concept of $w, \alpha, v, a$ 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.
$\frac{4g}{7}$
Q50 Allen 1. Concept of $w, \alpha, v, a$ 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.
$4m / s^{2}$