Conservation of Angular Momentum

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LEVELJEE Main

A solid sphere is rotating in free space. If the radius of the sphere is increased keeping mass same, which one of the following will not be affected?

Rotational kinetic energy

JEE Main 2021, 18 March Shift-I

LEVELJEE Main

A thin circular ring of mass $M$ and radius $r$ is rotating about its axis with an angular speed $ω$ . Two particles having mass $m$ each are now attached at diametrically opposite points. The angular speed of the ring will become

(A)

ω \frac{M}{M + m}

(B)

ω \frac{M + 2 m}{M}

(C)

ω \frac{M}{M + 2 m}

(D)

ω \frac{M - 2 m}{M + 2 m}

LEVELJEE Main

Initial angular velocity of a circular disc of mass $M$ is $ω_{1}$ . Then, two small spheres of mass $m$ are attached gently to two diametrically opposite points on the edge of the disc. What is the final angular velocity of the disc?

(A)

(\frac{M + m}{M}) ω_{1}

(B)

(\frac{M + m}{m}) ω_{1}

(C)

(\frac{M}{M + 4 m}) ω_{1}

(D)

(\frac{M}{M + 2 m}) ω_{1}

JEE Main 2020, 02 Sep Shift-II

LEVELJEE Main

Two uniform circular discs are rotating independently in the same direction around their common axis passing through their centres. The moment of inertia and angular velocity of the first disc are $0.1 kg-m^{2}$ and $10 rad s^{- 1}$ respectively, while those for the second one are $0.2 kg-m^{2}$ and $5 rad s^{- 1}$ , respectively. At some instant they get stuck together and start rotating as a single system about their common axis with some angular speed. The kinetic energy of the combined system is

(A)

\frac{10}{3} J

(B)

\frac{20}{3} J

(C)

\frac{5}{3} J

(D)

\frac{2}{3} J

LEVELJEE Main

A thin horizontal circular disc is rotating about a vertical axis passing through its centre. An insect is at rest at a point near the rim of the disc. The insect now moves along a diameter of the disc to reach its other end. During the journey of the insect, the angular speed of the disc

(A)

continuously decreases

(B)

continuously increases

(C)

first increases and then decreases

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, $ω_{1}$ and $ω_{2}$ respectively and are brought into contact face to face with their axes of rotation co-axial. The loss in kinetic energy of the system in the process is given by

(A)

\frac{I _{1} I _{2}}{( I _{1} + I _{2} )} (ω_{1} - ω_{2})^{2}

(B)

\frac{( I _{1} - I _{2} ) ^{2} ω _{1} ω _{2}}{2 ( I _{1} + I _{2} )}

(C)

\frac{I _{1} I _{2}}{2 ( I _{1} + I _{2} )} (ω_{1} - ω_{2})^{2}

(D)

\frac{( ω _{1} - ω _{2} ) ^{2}}{2 ( I _{1} + I _{2} )}

JEE Main 2020, 04 Sep Shift-I

LEVELJEE Main

A circular disc of mass $M$ and radius $R$ is rotating about its axis with angular speed $ω_{1}$ . If another stationary disc having radius $\frac{R}{2}$ and same mass $M$ is dropped co-axially on to the rotating disc. Gradually, both discs attain constant angular speed $ω_{2}$ . The energy lost in the process is $p %$ of the initial energy. Value of $p$ is ...........

JEE Main 2013

LEVELJEE Advanced

A hoop of radius $r$ and mass $m$ rotating with an angular velocity $ω_{0}$ is placed on a rough horizontal surface. The initial velocity of the centre of the hoop is zero. What will be the velocity of the centre of the hoop when it ceases to slip?

(A)

\frac{r ω _{0}}{4}

(B)

\frac{r ω _{0}}{3}

(C)

\frac{r ω _{0}}{2}

(D)

r ω_{0}

JEE Main 2020, 8 Jan Shift-I

LEVELJEE Advanced

Consider a uniform rod of mass $M = 4 m$ and length $l$ pivoted about its centre. A mass $m$ moving with velocity $v$ making angle $θ = \frac{π}{4}$ to the rod's long axis collides with one end of the rod and sticks to it. The angular speed of the rod-mass system just after the collision is

(A)

\frac{3 2}{7} \frac{v}{l}

(B)

\frac{4}{7} \frac{v}{l}

(C)

\frac{3}{7} \frac{v}{l}

(D)

\frac{3}{7 2} \frac{v}{l}

JEE Main 2020, 05 Sep Shift-I

LEVELJEE Main

A wheel is rotating freely with an angular speed $ω$ on a shaft. The moment of inertia of the wheel is $I$ and the moment of inertia of the shaft is negligible. Another wheel of moment of inertia $3 I$ initially at rest is suddenly coupled to the same shaft. The resultant fractional loss in the kinetic energy of the system is

(A)

\frac{3}{4}

(B)

\frac{5}{6}

(C)

0

(D)

\frac{1}{4}

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 2019, 10 April Shift-I

LEVELJEE Main

Two coaxial discs, having moments of inertia $I_{1}$ and $\frac{I _{1}}{2}$ are rotating with respective angular velocities $ω_{1}$ and $\frac{ω _{1}}{2}$ , about their common axis. They are brought in contact with each other and thereafter they rotate with a common angular velocity. If $E_{f}$ and $E_{i}$ are the final and initial total energies, then $(E_{f} - E_{i})$ is

(A)

- \frac{I _{1} ω _{1}^{2}}{24}

(B)

- \frac{I _{1} ω _{1}^{2}}{12}

(C)

\frac{3}{8} I_{1} ω_{1}^{2}

(D)

\frac{I _{1} ω _{1}^{2}}{6}

LEVELJEE Main

A thin circular ring of mass $m$ and radius $R$ is rotating about its axis with a constant angular velocity $ω$ . Two objects each of mass $M$ are attached gently to the opposite ends of a diameter of the ring. The ring now rotates with an angular velocity $ω^{'}$ =

(A)

\frac{ω ( m + 2 M )}{m}

(B)

\frac{ω ( m - 2 M )}{( m + 2 M )}

(C)

\frac{ω m}{( m + M )}

(D)

\frac{ω m}{( m + 2 M )}

JEE Main 2019, 9 April Shift-II

LEVELJEE Main

A thin smooth rod of length $L$ and mass $M$ is rotating freely with angular speed $ω_{0}$ about an axis perpendicular to the rod and passing through its centre. Two beads of mass $m$ and negligible size are at the centre of the rod initially. The beads are free to slide along the rod. The angular speed of the system, when the beads reach the opposite ends of the rod, will be

(A)

\frac{M ω _{0}}{M + 3 m}

(B)

\frac{M ω _{0}}{M + m}

(C)

\frac{M ω _{0}}{M + 2 m}

(D)

\frac{M ω _{0}}{M + 6 m}

JEE Main 2020, 03 Sep Shift-I

LEVELJEE Main

A person of 80 kg mass is standing on the rim of a circular platform of mass 200 kg rotating about its axis at 5 rpm. The person now starts moving towards the centre of the platform. What will be the rotational speed (in rpm) of the platform when the person reaches its centre?

JEE Advanced 2004

LEVELJEE Main

A child is standing with folded hands at the centre of a platform rotating about its central axis. The kinetic energy of the system is $K$ . The child now stretches his arms so that the moment of inertia of the system doubles. The kinetic energy of the system now is

(A)

2 K

(B)

K /2

(C)

K /4

(D)

4 K

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

JEE Advanced (2002)

LEVELJEE Main

A circular platform is free to rotate in a horizontal plane about a vertical axis passing through its centre. A tortoise is sitting at the edge of the platform. Now, the platform is given an angular velocity $ω_{0}$ . When the tortoise move along a chord of the platform with a constant velocity (with respect to the platform). The angular velocity of the platform $ω (t)$ will vary with time $t$ as

A rod of length $L$ and mass $M$ is hinged at point $O$ . A small bullet of mass $m$ hits the rod as shown in the figure. The bullet gets embedded in the rod. Find angular velocity of the system just after impact.

JEE Advanced 2011

LEVELJEE Main

A thin ring of mass $2 kg$ and radius $0.5 m$ is rolling without slipping on a horizontal plane with velocity $1 m/s$ . A small ball of mass $0.1 kg$ , moving with velocity $20 m/s$ in the opposite direction, hits the ring at a height of $0.75 m$ and goes vertically up with velocity $10 m/s$ . Immediately after the collision,

* Multiple Correct Options

(A)

the ring has pure rotation about its stationary CM

(B)

the ring comes to a complete stop

(C)

friction between the ring and the ground is to the left

(D)

there is no friction between the ring and the ground

JEE Advanced 1983

LEVELJEE Main

A thin circular ring of mass $M$ and radius $r$ is rotating about its axis with a constant angular velocity $ω$ . Two objects, each of mass $m$ , are attached gently to the opposite ends of a diameter of the ring. The wheel now rotates with an angular velocity

(A)

\frac{ω M}{M + m}

(B)

\frac{ω ( M - 2 m )}{M + 2 m}

(C)

\frac{ω M}{M + 2 m}

(D)

\frac{ω ( M + 2 m )}{M}

JEE Advanced 1999

LEVELJEE Main

A smooth sphere $A$ is moving on a frictionless horizontal plane with angular velocity $ω$ and centre of mass velocity $v$ . It collides elastically and head on with an identical sphere $B$ at rest. Neglect friction everywhere. After the collision their angular speeds are $ω_{A}$ and $ω_{B}$ respectively. Then,

(A)

ω_{A} < ω_{B}

(B)

ω_{A} = ω_{B}

(C)

ω_{A} = ω

(D)

ω_{B} = ω

JEE Advanced (1999)

LEVELJEE Main

A cubical block of side $a$ moving with velocity $v$ on a horizontal smooth plane as shown. It hits a ridge at point $O$ . The angular speed of the block after it hits $O$ is

(A)

\frac{3 v}{4 a}

(B)

\frac{3 v}{2 a}

(C)

\frac{3 v}{2 a}

An equilateral triangle $A B C$ formed from a uniform wire has two small identical beads initially located at $A$ . The triangle is set rotating about the vertical axis $A O$ . Then the beads are released from rest simultaneously and allowed to slide down, one along $A B$ and other along $A C$ as shown. Neglecting frictional effects, the quantities that are conserved as beads slides down are

(A)

(a) angular velocity and total energy (kinetic and potential)

(B)

(b) total angular momentum and total energy

(C)

(D)

(d) total angular momentum and moment of inertia about the axis of rotation

JEE Advanced (1994)

LEVELJEE Main

Two uniform rods $A$ and $B$ of length $0.6 m$ each and of masses $0.01 kg$ and $0.02 kg$ , respectively are rigidly joined end to end. The combination is pivoted at the lighter end, $P$ as shown in figure. Such that it can freely rotate about point $P$ in a vertical plane. A small object of mass $0.05 kg$ , moving horizontally, hits the lower end of the combination and sticks to it. What should be the velocity of the object, so that the system could just be raised to the horizontal position?

A solid sphere is rotating in free space. If the radius of the sphere is increased keeping mass same, which one of the following will not be affected?

A thin horizontal circular disc is rotating about a vertical axis passing through its centre. An insect is at rest at a point near the rim of the disc. The insect now moves along a diameter of the disc to reach its other end. During the journey of the insect, the angular speed of the disc

A person of 80 kg mass is standing on the rim of a circular platform of mass 200 kg rotating about its axis at 5 rpm. The person now starts moving towards the centre of the platform. What will be the rotational speed (in rpm) of the platform when the person reaches its centre?

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } A cubical block of side a moving with velocity v on a horizontal smooth plane as shown. It hits a ridge at point O. The angular speed of the block after it hits O is

A cubical block of side $a$ moving with velocity $v$ on a horizontal smooth plane as shown. It hits a ridge at point $O$ . The angular speed of the block after it hits $O$ is