Core Syllabus Module

Rotational Motion

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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 2012

LEVELJEE Advanced

Consider a disc rotating in the horizontal plane with a constant angular speed $ω$ about its centre $O$ . The disc has a shaded region on one side of the diameter and an unshaded region on the other side as shown in the figure. When the disc is in the orientation as shown, two pebbles $P$ and $Q$ are simultaneously projected at an angle towards $R$ . The velocity of projection is in the $y$ - $z$ plane and is same for both pebbles with respect to the disc. Assume that (i) they land back on the disc before the disc has completed $1/8$ rotation, (ii) their range is less than half the disc radius, and (iii) $ω$ remains constant throughout. Then

(A)

P

lands in the shaded region and

Q

in the unshaded region

(B)

P

lands in the unshaded region and

Q

in the shaded region

(C)

both

P

and

Q

land in the unshaded region

(D)

both

P

and

Q

land in the shaded region

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 2018

LEVELJEE Advanced

In the column-I below, four different paths of a particle are given as functions of time. In these functions, $α$ and $β$ are positive constants of appropriate dimensions and $α \neq = β$ . In each case, the force acting on the particle is either zero or conservative. In column-II, five physical quantities of the particle are mentioned: $p$ is the linear momentum, $L$ is the angular momentum about the origin, $K$ is the kinetic energy, $U$ is the potential energy and $E$ is the total energy. Match each path in Column-I with those quantities in Column-II, which are conserved for that path. $Column I A. r (t) = α t \hat{i} + β t \hat{j} B. r (t) = α cos ω t \hat{i} + β sin ω t \hat{j} C. r (t) = α (cos ω t \hat{i} + sin ω t \hat{j}) D. r (t) = α t \hat{i} + \frac{β}{2} t^{2} \hat{j} Column II p. p q. L r. K s. U t. E$

(A)

\to

p, q, r, s, t; B

\to

q, t; C

\to

q, r, s, t; D

\to

(B)

\to

p, q, r, s, t; B

\to

r, t; C

\to

q, r, s, t; D

\to

q, t

(C)

\to

q, r, s; B

\to

t; C

\to

p, q, s; D

\to

q, t

(D)

\to

p, q, r, t; B

\to

q, t; C

\to

q, r, s, t; D

\to

q, t

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 1999

LEVELJEE Advanced

A man pushes a cylinder of mass $m_{1}$ with the help of a plank of mass $m_{2}$ as shown. There is no slipping at any contact. The horizontal component of the force applied by the man is $F$ . Find (a) the accelerations of the plank and the centre of mass of the cylinder and (b) the magnitudes and directions of frictional forces at contact points.

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 Main 2019

LEVELJEE Advanced

A person of mass $M$ is sitting on a swing to length $L$ and swinging with an angular amplitude $θ_{0}$ . If the person stands up when the swing passes through its lowest point, the work done by him, assuming that his centre of mass moves by a distance $l (l ≪ L)$ , is close to

(A)

M g l (1 - θ_{0}^{2})

(B)

M g l (1 + θ_{0}^{2})

(C)

M g l

(D)

M g l (1 + \frac{θ _{0}^{2}}{2})

JEE Advanced 2019

LEVELJEE Advanced

A thin and uniform rod of mass $M$ and length $L$ is held vertical on a floor with large friction. The rod is released from rest so that it falls by rotating about its contact-point with the floor without slipping. Which of the following statement(s) is/are correct, when the rod makes an angle $6 0^{\circ}$ with vertical ? [g is the acceleration due to gravity]

* Multiple Correct Options

(A)

The radial acceleration of the rod's center of mass will be

\frac{3 g}{4}

(B)

The angular acceleration of the rod will be

\frac{2 g}{L}

(C)

The angular speed of the rod will be

\frac{3 g}{2 L}

(D)

The normal reaction force from the floor on the rod will be

\frac{M g}{16}

JEE Advanced 2026

LEVELJEE Advanced

A solid cylinder of radius $R$ rolls without slipping with a center of mass speed $v_{0} = \frac{g R}{3}$ on a horizontal surface with a vertical edge, as shown in the figure. Here, $g$ is the acceleration due to the gravity. At the moment when the cylinder loses contact with the surface due to rotation around the corner, the speed of its center of mass is:

(A)

0

(B)

\frac{5 g R}{7}

(C)

\frac{g R}{15}

(D)

\frac{3 g R}{7}

JEE Advanced 2026

LEVELJEE Advanced

Comprehension Passage

A uniform circular disk of radius

0.2 m

and mass

1 kg

is pivoted at its top point

C

such that it can rotate freely around

C

in the

X Y

plane, as shown in the figure. Initially, when the disk is at rest, a particle of mass

20 g

, travelling along negative

x

direction in the

X Y

plane with speed

100 ms^{- 1}

, hits the circumference of the disk at a point

P

. After collision the particle moves along negative

y

direction at a speed of

90 ms^{- 1}

. [Given: the acceleration due to gravity

(g) = - 10 \hat{j} ms^{- 2}

]

Question 1:

After the collision the disk starts to rotate around point $C$ in the $X Y$ plane. The maximum change in the height (in m) of its center $O$ is:

Question 2:

Amount of energy loss (in J) in the collision is:

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 2020, 3 Sep Shift-I

LEVELJEE Advanced

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 $θ$ before momentarily coming to rest. If the rod has mass $M = 2 kg$ and length $l = 1 m$ , then the value of $θ$ is approximately (Take, $g = 10 m/s^{2}$ )

(A)

6 3^{\circ}

(B)

5 5^{\circ}

(C)

6 9^{\circ}

(D)

4 9^{\circ}

JEE Main 2020

LEVELJEE Advanced

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 $\frac{N I _{0}}{16}$ , where N is an integer. Value of N is ……… .

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 (2012)

LEVELJEE Advanced

A lamina is made by removing a small disc of diameter 2R from a bigger disc of uniform mass density and radius 2R, as shown in the figure. The moment of inertia of this lamina about axes passing through O and P is $I_{O}$ and $I_{P}$ , respectively. Both these axes are perpendicular to the plane of the lamina. The ratio $\frac{I _{P}}{I _{O}}$ to the nearest integer is

JEE Advanced 2015

LEVELJEE Advanced

A ring of mass $M$ and radius $R$ is rotating with angular speed $ω$ about a fixed vertical axis passing through its centre $O$ with two point masses each of mass $\frac{M}{8}$ at rest at $O$ . These masses can move radially outwards along two massless rods fixed on the ring as shown in the figure. At some instant, the angular speed of the system is $\frac{8}{9} ω$ and one of the masses is at a distance of $\frac{3}{5} R$ from $O$ . At this instant, the distance of the other mass from $O$ is

(A)

\frac{2}{3} R

(B)

\frac{1}{3} R

(C)

\frac{3}{5} R

(D)

\frac{4}{5} R

Micro-Topic Strategy Matrix

Kinematics of Rotational Motion

Moment of Inertia

Torque and Angular Momentum

Dynamics of Rigid Body Rotation

Work and Energy in Rotational Motion

Rolling Motion

Conservation of Angular Momentum

Chapter Elite-20 Challenge Zone

Comprehension Passage

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } After the collision the disk starts to rotate around point C in the XY plane. The maximum change in the height (in m) of its center O is:

Amount of energy loss (in J) in the collision is:

After the collision the disk starts to rotate around point $C$ in the $X Y$ plane. The maximum change in the height (in m) of its center $O$ is: