Dynamics of Rigid Body Rotation

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JEE Main 2019, 11 Jan Shift-II

A string is wound around a hollow cylinder of mass 5 kg and radius 0.5 m. If the string is now pulled with a horizontal force of 40 N and the cylinder is rolling without slipping on a horizontal surface (see figure), then the angular acceleration of the cylinder will be (Neglect the mass and thickness of the string)

^{2}

(B)

16 rad /s

^{2}

(C)

20 rad /s

^{2}

(D)

12 rad /s

^{2}

JEE Main 2019, 12 April Shift-I

LEVELJEE Main

A uniform rod of length $l$ is being rotated in a horizontal plane with a constant angular speed about an axis passing through one of its ends. If the tension generated in the rod due to rotation is $T (x)$ at a distance $x$ from the axis, then which of the following graphs depicts it most closely?

JEE Main 2019, 9 April Shift-I

LEVELJEE Main

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 $θ$ , where $θ$ is the angle by which it has rotated, is given as $k θ^{2}$ . If its moment of inertia is $I$ , then the angular acceleration of the disc is

(A)

\frac{k}{2 I} θ

(B)

\frac{k}{I} θ

(C)

\frac{k}{4 I} θ

(D)

\frac{2 k}{I} θ

JEE Main 2019

LEVELJEE Main

A rigid massless rod of length $3 l$ has two masses attached at each end as shown in the figure. The rod is pivoted at point $P$ on the horizontal axis (see figure). When released from initial horizontal position, its instantaneous angular acceleration will be

(A)

\frac{g}{13 l}

(B)

\frac{g}{2 l}

(C)

\frac{7 g}{3 l}

(D)

\frac{g}{3 l}

JEE Main 2017

LEVELJEE Main

A slender uniform rod of mass $M$ and length $l$ is pivoted at one end so that it can rotate in a vertical plane (see the figure). There is negligible friction at the pivot. The free end is held vertically above the pivot and then released. The angular acceleration of the rod when it makes an angle $θ$ with the vertical, is

(A)

\frac{2 g}{3 l} sin θ

(B)

\frac{3 g}{2 l} cos θ

(C)

\frac{2 g}{3 l} cos θ

(D)

\frac{3 g}{2 l} sin θ

JEE Main 2019, 10 Jan Shift-I

LEVELJEE Advanced

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 $μ$ , the torque applied by the machine on the mop in (N-m) is

(A)

\frac{2}{3} μ F R

(B)

\frac{μ F R}{6}

(C)

\frac{μ F R}{3}

(D)

\frac{μ F R}{2}

JEE Main 2020, 03 Sep Shift-II

LEVELJEE Advanced

A uniform rod of length $l$ is pivoted at one of its ends on a vertical shaft of negligible radius. When the shaft rotates at angular speed $ω$ , the rod makes an angle $θ$ with it (see figure). To find $θ$ , equate the rate of change of angular momentum (direction going into the paper) $\frac{m l ^{2}}{12} ω^{2} sin θ cos θ$ about the centre of mass to the torque provided by the horizontal and vertical forces $F_{H}$ and $F_{V}$ about the centre of mass. The value of $θ$ is then such that

(A)

cos θ = \frac{g}{l ω ^{2}}

(B)

cos θ = \frac{g}{2 l ω ^{2}}

(C)

cos θ = \frac{2 g}{3 l ω ^{2}}

(D)

cos θ = \frac{3 g}{2 l ω ^{2}}

JEE Main 2014

LEVELJEE Main

A mass $m$ 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 acceleraton will the mass fall on release?

(A)

\frac{2 g}{3}

(B)

\frac{g}{2}

(C)

\frac{5 g}{6}

(D)

g

LEVELJEE Main

A mass $m$ hangs with the help of a string wrapped around a pulley on a frictionless bearing. The pulley has mass $m$ and radius $R$ . Assuming pulley to be a perfect uniform circular disc, the acceleration of the mass $m$ , if the string does not slip on the pulley, is

(A)

g

(B)

\frac{2}{3} g

(C)

\frac{g}{3}

(D)

\frac{3}{2} g

JEE Main 2019, 8 April Shift-II

LEVELJEE Advanced

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 $5 m$ . When released, it slips off the table in a very short time $τ = 0.01 s$ , remaining essentially horizontal. The angle by which it would rotate when it hits the ground will be (in radians) close to

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 $μ = 0.4$ , the maximum possible value of $100 \times \frac{b}{a}$ for box not to topple before moving is ……… .

JEE Advanced 2009

LEVELJEE Advanced

A block of base $10 cm \times 10 cm$ and height $15 cm$ is kept on an inclined plane. The coefficient of friction between them is $3$ . The inclination $θ$ of this inclined plane from the horizontal plane is gradually increased from $0^{\circ}$ . Then,

(A)

θ = 3 0^{\circ}

, the block will start sliding down the plane

(B)

the block will remain at rest on the plane up to certain

θ

and then it will topple

(C)

θ = 6 0^{\circ}

, the block will start sliding down the plane and continue to do so at higher angles

(D)

θ = 6 0^{\circ}

, the block will start sliding down the plane and on further increasing

θ

, it will topple at certain

θ

JEE Advanced 2000

LEVELJEE Advanced

A rod $A B$ of mass $M$ and length $L$ is lying on a horizontal frictionless surface. A particle of mass $m$ travelling along the surface hits the end $A$ of the rod with a velocity $v_{0}$ in a direction perpendicular to $A B$ . The collision is elastic. After the collision, the particle comes to rest. (a) Find the ratio $m / M$ . (b) A point $P$ on the rod is at rest immediately after collision. Find the distance $A P$ . (c) Find the linear speed of the point $P$ a time $π L /3 v_{0}$ after the collision.

JEE Advanced 1995

LEVELJEE Advanced

A rectangular rigid fixed block has a long horizontal edge. A solid homogeneous cylinder of radius $R$ is placed horizontally at rest with its length parallel to the edge such that the axis of the cylinder and the edge of the block are in the same vertical plane as shown in figure. There is sufficient friction present at the edge, so that a very small displacement causes the cylinder to roll off the edge without slipping. Determine (a) the angle $θ_{c}$ through which the cylinder rotates before it leaves contact with the edge, (b) the speed of the centre of mass of the cylinder before leaving contact with the edge and (c) the ratio of the translational to rotational kinetic energies of the cylinder when its centre of mass is in horizontal line with the edge.

JEE Advanced 2000

LEVELJEE Main

A cubical block of side $L$ rests on a rough horizontal surface with coefficient of friction $μ$ . 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

(A)

infinitesimal

(B)

\frac{m g}{4}

(C)

\frac{m g}{2}

(D)

m g (1 - μ)

JEE Advanced 2001

LEVELJEE Advanced

Two heavy metallic plates are joined together at $9 0^{\circ}$ to each other. A laminar sheet of mass $30 kg$ is hinged at the line $A B$ joining the two heavy metallic plates. The hinges are frictionless. The moment of inertia of the laminar sheet about an axis parallel to $A B$ and passing through its centre of mass is $1.2 kg-m^{2}$ . Two rubber obstacles $P$ and $Q$ are fixed, one on each metallic plate at a distance $0.5 m$ from the line $A B$ . This distance is chosen, so that the reaction due to the hinges on the laminar sheet is zero during the impact. Initially the laminar sheet hits one of the obstacles with an angular velocity $1 rad/s$ and turns back. If the impulse on the sheet due to each obstacle is $6 N-s$ . (a) Find the location of the centre of mass of the laminar sheet from $A B$ . (b) At what angular velocity does the laminar sheet come back after the first impact ? (c) After how many impacts, does the laminar sheet come to rest ?

JEE Advanced 1984

LEVELJEE Main

A uniform cube of side $a$ and mass $m$ rests on a rough horizontal table. A horizontal force $F$ is applied normal to one of the faces at a point that is directly above the centre of the face, at a height $3 a /4$ above the base. The minimum value of $F$ for which the cube begins to tip about the edge is ......... (Assume that the cube does not slide).

JEE Advanced 1997

LEVELJEE Main

A rod of weight $W$ is supported by two parallel knife edges $A$ 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 distance $x$ from $A$ . The normal reaction on $A$ is ......... and on $B$ is ......... .

JEE Advanced (2016)

LEVELOlympiad

Two thin circular discs of mass $m$ and $4 m$ , having radii of $a$ and $2 a$ , respectively, are rigidly fixed by a massless, rigid rod of length $l = 24 a$ through their centers. This assembly is laid on a firm and flat surface and set rolling without slipping on the surface so that the angular speed about the axis of the rod is $ω$ . The angular momentum of the entire assembly about the point ' $O$ ' is $L$ (see the figure). Which of the following statement(s) is (are) true?

* Multiple Correct Options

(A)

The magnitude of the z-component of

L

55 m a^{2} ω

(B)

The magnitude of angular momentum of centre of mass of the assembly about the point

O

81 m a^{2} ω

(C)

The centre of mass of the assembly rotates about the Z-axis with an angular speed of

\frac{ω}{5}

(D)

The magnitude of angular momentum of the assembly about its centre of mass is

17 m a^{2} \frac{ω}{2}

JEE Advanced 2002

LEVELJEE Main

Three particles $A$ , $B$ and $C$ , each of mass $m$ , are connected toeach other by three massless rigid rods to form a rigid , equilateral triangular body of side $l$ . This body is placed on a horizontal frictionless table ( $x$ - $y$ plane) and is hinged to it at the point $A$ , so that it can move without friction about the vertical axis through $A$ (see figure). The body is set into rotational motion on the table about $A$ with a constant angular velocity $ω$ . (a) Find the magnitude of the horizontal force exerted by the hinge on the body. (b) At time $T$ , when the side $B C$ is parallel to the $X$ -axis, a force $F$ is applied on $B$ along $B C$ (as shown). Obtain the $x$ -component and the $y$ -component of the force exerted by the hinge on the body, immediately after time $T$ .

JEE Advanced 2011

LEVELJEE Advanced

A boy is pushing a ring of mass $2 kg$ and radius $0.5 m$ with a stick as shown in the figure. The stick applies a force of $2 N$ on the ring and rolls it without slipping with an acceleration of $0.3 m/s^{2}$ . The coefficient of friction between the ground and the ring is large enough that rolling always occurs and the coefficient of friction between the stick and the ring is $(\frac{P}{10})$ . The value of $P$ is

JEE Advanced 1995

LEVELJEE Advanced

A rectangular rigid fixed block has a long horizontal edge. A solid homogeneous cylinder of radius $R$ is placed horizontally at rest with its length parallel to the edge such that the axis of the cylinder and the edge of the block are in the same vertical plane as shown in figure. There is sufficient friction present at the edge, so that a very small displacement causes the cylinder to roll off the edge without slipping. Determine (a) the angle $θ_{c}$ through which the cylinder rotates before it leaves contact with the edge, (b) the speed of the centre of mass of the cylinder before leaving contact with the edge and (c) the ratio of the translational to rotational kinetic energies of the cylinder when its centre of mass is in horizontal line with the edge.

JEE Advanced 2007

LEVELJEE Main

Assertion If there is no external torque on a body about its centre of mass, then the velocity of the centre of mass remains constant. Reason The linear momentum of an isolated system remains constant.

(A)

If Assertion is true, Reason is true; Reason is the correct explanation for Assertion

(B)

If Assertion is true, Reason is true; Reason is not a correct explanation for Assertion

(C)

If Assertion is true; Reason is false

(D)

If Assertion is false; Reason is true

JEE Advanced 2022

LEVELJEE Advanced

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 $ω$ about the vertical axis passing through the center of $A$ . The angular momentum of $B$ is $n M ω R^{2}$ with respect to the center of $A$ . Which of the following is the value of $n$ ?

(A)

(B)

(C)

\frac{7}{2}

(D)

\frac{9}{2}

JEE Advanced 2023

LEVELJEE Advanced

A thin circular coin of mass $5 gm$ and radius $4/3 cm$ is initially in a horizontal xy-plane. The coin is tossed vertically up ( $+ z$ direction) by applying an impulse of $\frac{π}{2} \times 1 0^{- 2} N-s$ at a distance $2/3 cm$ from its center. The coin spins about its diameter and moves along the $+ z$ direction. By the time the coin reaches back to its initial position, it completes $n$ rotations. The value of $n$ is ____. [Given: The acceleration due to gravity $g = 10 ms^{- 2}$ ]