Angular Momentum

Q.

Steering of vehicles is an example of________. (wheel and axle/screw).

Q. Two small particles $A$ and $B$ of masses $M$ and $2M$ respectively, are joined rigidly to the ends of a light rod of length $l$ as shown in the figure. The system translates on a smooth horizontal surface with a velocity $u$ in a direction perpendicular to the rod. A particle $P$ of mass $M$ kept at rest on the surface sticks to the particle $A$ as the particle $P$ collides with it. Then choose the correct option(s).

1. The speed of particle $A$ just after collision is $\frac{u}{2}$
2. The speed of particle $B$ just after collision is $\frac{u}{2}$
3. The speed of particle $B$ just after collision is $u$
4. The angular velocity of the system $(A+B+P+\text{rod})$ just after collision is $\frac{u}{2l}$

Q.

A small meteorite of mass 'm' travelling towards the centre of earth strikes the earth at the equator. The earth is a uniform sphere of mass 'M' and radius 'R'. The length of the day was 'T' before the meteorite struck. After the meteorite strikes the earth, the length of day increases (in sec) by

1. $\frac{5mT}{2M}$
2. $\frac{m}{M}T$
3. $\frac{4mT}{5M}$
4. $\frac{M}{3mT}$

Q.

A uniform rod of mass M and length ℓ is placed on a smooth horizontal surface with its one end pivoted to the surface. A small ball of mass m moving along the surface with a velocity v₀, perpendicular to the rod, collides elastically with the free end of the rod. Find the impulse applied by the pivot on the rod during collision. (Take M/m = 2)

1. $\frac{1}{5m{v}_0}$

2. $\frac{3}{5m{v}_0}$

3. $\frac{2}{5m{v}_0}$

4. None of these

Q. Three partricles $A,B$ and $C$ of different masses $1\text{kg},2\text{kg}$ and $3\text{kg}$ are moving with different velocities $v_A=\hat{i}+2\hat{j},v_B=2\hat{i}+4\hat{j}$ and $v_C=3\hat{j}$. At time $t$, particles $A,B$ & $C$ are at points $(2,3),(3,2)$ & $(1,4)$ respectively. The total angular momentum of the system of $3$ particles about the origin [in ${\text{kg m}}^2\text{/s}$] is

1. $26\hat{k}$
2. $9\hat{k}$
3. $10\hat{k}$
4. $5\hat{k}$

Q. A particle of mass $2\text{kg}$ is projected on horizontal ground with an initial velocity $u=20\text{m/s}$ making an angle ${60}^{\circ }$ with the vertical. Find out the angular momentum of the particle about the point of projection when it just strikes the ground. [in ${\text{kg m}}^2\text{/s}$]

1. $200\sqrt{3}$
2. $100\sqrt{3}$
3. $\text{zero}$
4. $400\sqrt{3}$

Q. A boy sitting firmly over a rotating stool has his arms folded. If he stretches his arms, his angular momentum about the axis of rotation

1. increases
2. decreases
3. remains unchanged
4. Can't say

Q. A particle of mass $1\text{kg}$ is projected with a velocity $20\sqrt{2}\text{m/s}$ from the origin of an $x-y$ co- ordinate axis system at an angle ${45}^{\circ }$ with $x$ axis (horizontal). The angular momentum [in ${\text{kg m}}^2\text{/s}$] of the ball about the point of projection after $1\text{second}$ of projection is [Take $g=10{\text{m/s}}^2$ and $y$ - axis is taken as vertical]

1. $200\hat{k}$
2. $-100\hat{k}$
3. $300\hat{j}$
4. $-350\hat{j}$

Q. A particle is moving in x-y plane. Which component of linear momentum does not contribute to angular momentum?

1. Horizontal component
2. Tangential component
3. Radial component
4. Vertical component

Q. A particle of mass $m=1\text{kg}$ is moving along the line $y=x+2$ with speed $2\text{m/s}$. The angular momentum of the particle about origin $\text{O}$ is

1. $\overrightarrow{L}=0$
2. $\overrightarrow{L}=2\sqrt{2}\left(\hat{k}\right)$
3. $\overrightarrow{L}=2\sqrt{2}(-\hat{k})$
4. $\overrightarrow{L}=\sqrt{2}(-\hat{k})$

Q. A particle of mass $200\text{g}$ is projected from origin $O$ with speed $10\text{m/s}$ at an angle ${60}^{\circ }$ with positive $x-$ axis. Find the magnitude of angular momentum of particle after $1\text{sec}$ about the origin. (Take $g=10{\text{m/s}}^2$)

1. $25{\text{kg-m}}^2/\text{sec}$
2. $0{\text{kg-m}}^2/\text{sec}$
3. $250{\text{kg-m}}^2/\text{sec}$
4. $125{\text{kg-m}}^2/\text{sec}$

Q. A particle having mass $2\text{kg}$ is moving along a straight line $3x+4y=5$ with speed $8\text{m/s}$. Find angular momentum of the particle about the origin.

1. $20\text{kg}\frac{{\text{m}}^2}{\text{s}}$
2. Zero
3. $16\text{kg}\frac{{\text{m}}^2}{\text{s}}$
4. $\frac{80}{3}\text{kg}\frac{{\text{m}}^2}{\text{s}}$

Q. A particle of mass $2\text{kg}$ is projected on horizontal ground with an initial velocity $u=20\text{m/s}$ making an angle ${60}^{\circ }$ with the vertical. Find out the angular momentum of the particle about the point of projection when it just strikes the ground. [in ${\text{kg m}}^2\text{/s}$]

1. $200\sqrt{3}$
2. $100\sqrt{3}$
3. $\text{zero}$
4. $400\sqrt{3}$

Q.

A uniform bar of length 6a and mass 8 m lies on a smooth horizontal table. Two point masses m and 2m moving in the same horizontal plane with speed 2v and v respectively, strike the bar and stick to the bar after collision. Denoting angular velocity (about the centre of mass), total energy and centre of mass velocity by $\omega$, $E$ and $v_c$ respectively, we have after collision

1. $v_c=0$

2. $\omega =\frac{3v}{5a}$

3. $\omega =\frac{v}{5a}$

4. $E=\frac{3}{5}m{v}^2$

Q. A particle of mass $m$ is moving with a constant velocity $v$ parallel to the $x-$ axis as shown in the figure. Its angular momentum (magnitude) about origin $\text{O}$ is

1. $mvb$
2. $mva$
3. $mv\sqrt{a^2+b^2}$
4. $mv(a+b)$

Q. A uniform rod having mass $m$ and length $L$ lies on a horizontal surface. A particle of mass $\frac{m}{2}$ moving with speed $u$ strikes the rod perpendicularly at one end and sticks to the rod as shown in the figure. Find the total angular momentum of the system about the origin just after the collision, if the angular velocity of the rod just after the collision is $\omega$ about its centre.

1. $[\frac{muL}{3}+\frac{m{L}^2\omega }{2}](-\hat{k})$
2. $[\frac{muL}{3}-\frac{m{L}^2\omega }{2}](-\hat{k})$
3. Zero
4. $[\frac{-muL}{3}-m{L}^2\omega ](-\hat{k})$

Q. A circular platform is free to rotate in a horizontal plane about a vertical axis passing through its center. A tortoise is sitting at the edge of the platform. Now, the platform is given an angular velocity ${\omega }_0$. When the tortoise moves along a chord of the platform with a constant velocity (with respect to the platform), the angular velocity of the platform $\omega$(t) will vary with time t as

1. 
2. 
3. 
4. 

Q. Binary star system consists of $2$ stars such that mass of one star is twice that of another. Which of the statement is correct?

1. One of the stars has twice the angular momentum about common COM as compared to the other one
2. Both have same angular momentum about centre of mass
3. Both stars have same linear speed
4. Both have same kinetic energy

Q. A kid of mass $50\text{kg}$ stands at the edge of a platform of radius $1\text{m}$ which can be freely rotated about its axis. The moment of inertia of the platform is $0.02{\text{kg-m}}^2$. The system is at rest. The kid throws a ball of mass $2\text{kg}$ horizontally to his friend (friend standing on the ground) in a direction tangential to the rim with a speed $10\text{m/s}$ as seen by his friend. The angular velocity with which the platform will start rotating in $\text{rad/s}$ is

1. $0.2\text{rad/s}$
2. $0.4\text{rad/s}$
3. $0.6\text{rad/s}$
4. $0.8\text{rad/s}$

Q. A particle having mass $m=2\text{kg}$ is moving with velocity $(2\hat{i}+3\hat{j})\text{m/s}$. Find angular momentum of the particle about the origin when it is at $(1,1,0)$ [in ${\text{kg m}}^2\text{/s}$].

1. $2\hat{k}$
2. $4\hat{k}$
3. $6\hat{k}$
4. Zero

Q. In the above problem the angular velocity of the system after the particle sticks to it will be

1. 0.3 rad/s
2. 5.3 rad/s
3. 10.3 rad/s
4. 89.3 rad/s

Q. A body of mass $2\text{kg}$ is projected with a speed $4\text{m/s}$ at an angle ${45}^{\circ }$ to the horizontal. What will be its angular momentum about the point of projection when the body is at the highest point in its trajectory?

1. $\frac{1.6}{\sqrt{2}}{\text{kg m}}^2/\text{s}$
2. $\frac{32.6}{\sqrt{2}}{\text{kg m}}^2/\text{s}$
3. $\frac{16}{\sqrt{2}}{\text{kg m}}^2/\text{s}$
4. $\frac{3.26}{\sqrt{2}}{\text{kg m}}^2/\text{s}$

Q. A uniform solid hemisphere of radius 10 m and mass 64 kg is placed with its curved surface on the smooth horizontal surface and a string AB of length 4 m is attached to point A on its rim as shown in the figure. Find the tension in the string if the hemisphere is in equilibrium.

1. 320 N
2. 640 N
3. 180 N
4. 360 N

Q. A particle of mass $m=2\text{kg}$ starts moving on a straight line $y=2$ with a constant velocity $10\hat{i}\text{m/s}$. Its angular momentum about origin $\left(\overrightarrow{L}\right)$ for the path from point $A$ to point $B$

1. Continuously decreases
2. Continuously increases
3. Remains constant
4. First increases and then decreases.

Q. A uniform rod of mass $M$ and length $2\text{m}$ lies on a smooth horizontal table. A particle of mass $m$ moving on the table with a velocity $u$ strikes the rod perpendicularly at an end and stops there. Find the distance travelled by the center of the rod (in metres) by the time it turns through a right angle. Given $M=4m$.

1. $\pi /12$
2. $\pi /6$
3. $\pi /3$
4. $\pi$

Q. A particle is at rest at a point $(4,3,0)$. Due to an impulse, it gets linear momentum $\overrightarrow{P}=\hat{i}+2\hat{j}$. The angular momentum of the particle about the origin is

1. $11(-\hat{k})$
2. $5(-\hat{k})$
3. $5\hat{k}$
4. $11\left(\hat{k}\right)$

Q. Two particles of same mass $2\text{kg}$ are rigidly attached to the ends of a massless rod of length $10\text{m}$. The rod is clamped at the centre such that the system rotates about the perpendicular bisector of the rod at an angular speed $10\text{rad/s}$. Calculate the magnitude of angular momentum of the system about the axis of rotation [in ${\text{kg m}}^2\text{/s}$]

1. $100$
2. $1000$
3. $500$
4. $10$

Q. A particle moves with a constant velocity parallel to the $Y$ axis. Its angular momentum about the origin

1. is zero
2. remains constant
3. goes on increasing
4. goes on decreasing

Q. A small mass $m$ is attached to a massless string whose other end is fixed at $\text{P}$ as shown in the figure. The mass is undergoing circular motion in the x-y plane with centre at $\text{O}$ and constant angular speed $\omega$. If the angular momentum of the system, calculated about $\text{O}$ and $\text{P}$ are denoted by ${\overrightarrow{L}}_O$ and ${\overrightarrow{L}}_P$ respectively, then

1. ${\overrightarrow{L}}_O$ and ${\overrightarrow{L}}_P$ do not vary with time.
2. ${\overrightarrow{L}}_O$ varies with time while ${\overrightarrow{L}}_P$ remains constant.
3. ${\overrightarrow{L}}_O$ remains constant while ${\overrightarrow{L}}_P$ varies with time.
4. ${\overrightarrow{L}}_O$ and ${\overrightarrow{L}}_P$ both vary with time.

Q. A solid sphere is set to rotate in the anticlockwise direction, on a rough horizontal surface with a linear speed $10\text{m/s}$ and an angular speed $5\text{rad/s}$ as shown in figure. Find the linear speed of the sphere when it stops rotating.

1. $6\text{m/s}$
2. $10\text{m/s}$
3. $30\text{m/s}$
4. Zero