Work and Energy in Rotational Motion

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JEE Main 2021 (27 July Shift-II)

In the given figure, two wheels $P$ and $Q$ are connected by a belt $B$ . The radius of $P$ is three times as that of $Q$ . In case of same rotational kinetic energy, the ratio of rotational inertias $(\frac{I _{1}}{I _{2}})$ will be $x : 1$ . The value of $x$ will be ………… .

JEE Main 2020, 9 Jan Shift-II

A uniformly thick wheel with moment of inertia $I$ and radius $R$ is free to rotate about its centre of mass (see figure). A massless string is wrapped over its rim and two blocks of masses $m_{1}$ and $m_{2} (m_{1} > m_{2})$ are attached to the ends of the string. The system is released from rest. The angular speed of the wheel when $m_{1}$ descents by a distance $h$ is

[\frac{2 ( m _{1} + m _{2} ) g h}{( m _{1} + m _{2} ) R ^{2} + I}]^{1/2}

(B)

[\frac{m _{1} + m _{2}}{( m _{1} + m _{2} ) R ^{2} + I} g h]^{1/2}

(C)

[\frac{m _{1} - m _{2}}{( m _{1} + m _{2} ) R ^{2} + I} g h]^{1/2}

(D)

[\frac{2 ( m _{1} - m _{2} ) g h}{( m _{1} + m _{2} ) R ^{2} + I}]^{1/2}

LEVELJEE Main

A thin uniform rod of length $l$ and mass $m$ is swinging freely about a horizontal axis passing through its end. Its maximum angular speed is $ω$ . Its centre of mass rises to a maximum height of

(A)

\frac{1}{3} \frac{l ^{2} ω ^{2}}{g}

(B)

\frac{1}{6} \frac{l ω}{g}

(C)

\frac{1}{2} \frac{l ^{2} ω ^{2}}{g}

(D)

\frac{1}{6} \frac{l ^{2} ω ^{2}}{g}

JEE Main 2021, 26 Feb Shift-II

LEVELJEE Main

A cord is wound round the circumference of wheel of radius $r$ . The axis of the wheel is horizontal and the moment of inertia about it is $I$ . A weight $m g$ is attached to the cord at the end. The weight falls from rest. After falling through a distance $h$ , the square of angular velocity of wheel will be

(A)

\frac{2 m g h}{I + 2 m r ^{2}}

(B)

\frac{2 m g h}{I + m r ^{2}}

(C)

2 g h

(D)

\frac{2 g h}{I + m r ^{2}}

JEE Main 2021, 31 Aug Shift-I

LEVELJEE Main

A 2 kg steel rod of length 0.6 m is clamped on a table vertically at its lower end and is free to rotate in vertical plane. The upper end is pushed so that the rod falls under gravity. Ignoring the friction due to clamping at its lower end, the speed of the free end of rod when it passes through its lowest position is ........... $ms^{- 1}$ . (Take, $g = 10 ms^{- 2}$ )

JEE Main 2019, 9 April Shift-II

LEVELJEE Main

Moment of inertia of a body about a given axis is $1.5 kg m^{2}$ . Initially, the body is at rest. In order to produce a rotational kinetic energy of $1200 J$ , the angular acceleration of $20 rad/s^{2}$ must be applied about the axis for a duration of

(A)

5 s

(B)

2 s

(C)

3 s

(D)

2.5 s

JEE Main 2019

LEVELJEE Main

A rod of length $50 cm$ is pivoted at one end. It is raised such that it makes an angle of $3 0^{\circ}$ from the horizontal as shown and released from rest. Its angular speed when it passes through the horizontal (in $rad s^{- 1}$ ) will be (Take, $g = 10 ms^{- 2}$ )

(A)

\frac{30}{2}

(B)

30

(C)

\frac{20}{3}

(D)

\frac{30}{2}

JEE Main 2020, 9 Jan Shift-I

LEVELJEE Main

One end of a straight uniform 1 m long bar is pivoted on horizontal table. It is released from rest when it makes an angle 30º from the horizontal (see figure). Its angular speed when its hits the table is given as $n s^{- 1}$ , where $n$ is an integer. The value of $n$ is ......

JEE Main 2020

LEVELJEE Main

As shown in the figure, a bob of mass $m$ is tied by a massless string whose other end portion is wound on a flywheel (disc) of radius $r$ and mass $m$ . When released from rest the bob starts falling vertically. When it has covered a distance of $h$ , the angular speed of the wheel will be

(A)

\frac{1}{r} \frac{2 g h}{3}

(B)

r \frac{3}{4 g h}

(C)

\frac{1}{r} \frac{4 g h}{3}

(D)

r \frac{3}{2 g h}

JEE Advanced (1990)

LEVELJEE Main

A carpet of mass $M$ made of inextensible material is rolled along its length in the form of a cylinder of radius $R$ and is kept on a rough floor. The carpet starts unrolling without sliding on the floor when a negligibly small push is given to it. Calculate the horizontal velocity of the axis of the cylindrical part of the carpet when its radius reduces to $R /2$ .

JEE Advanced 1987

LEVELJEE Main

A small sphere rolls down without slipping from the top of a track in a vertical plane. The track has an elevated section and a horizontal part. The horizontal part is $1.0 m$ above the ground level and the top of the track is $2.6 m$ above the ground. Find the distance on the ground with respect to the point $B$ (which is vertically below the end of the track as shown in figure) where the sphere lands. During its flight as a projectile, does the sphere continue to rotate about its centre of mass? Explain.

JEE Advanced 1998

LEVELJEE Main

A uniform circular disc has radius $R$ and mass $m$ . A particle, also of mass $m$ , is fixed at a point $A$ on the edge of the disc as shown in the figure. The disc can rotate freely about a horizontal chord $P Q$ that is at a distance $R /4$ from the centre $C$ of the disc. The line $A C$ is perpendicular to $P Q$ . Initially the disc is held vertical with the point $A$ at its highest position. It is then allowed to fall, so that it starts rotation about $P Q$ . Find the linear speed of the particle as it reaches its lowest position.

JEE Advanced 2015

LEVELJEE Advanced

Two identical uniform discs roll without slipping on two different surfaces $A B$ and $C D$ (see figure) starting at $A$ and $C$ with linear speeds $v_{1}$ and $v_{2}$ , respectively, and always remain in contact with the surfaces. If they reach $B$ and $D$ with the same linear speed and $v_{1} = 3 m/s$ , then $v_{2}$ in m/s is (Take $g = 10 m/s^{2}$ )