Angular Momentum

Q. A particle of mass $m=5\text{kg}$ is moving with a uniform speed $3\sqrt{2}\text{m/s}$ in $XY$ plane along the line $y=x+4$. The angular momentum about the origin is
[in ${\text{kg m}}^2\text{/s}$]

1. zero
2. $20$
3. $40$
4. $60$

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. Two rotating bodies $A$ and $B$ of masses $m$ and $2m$ with moments of inertia $I_A$ and $I_B(I_B>I_A)$ have equal kinetic energy of rotation. If $L_A$ and $L_B$ be their angular momenta respectively, then

1. $L_A=\frac{L_B}{2}$
2. $L_A=2L_B$
3. $L_B>L_A$
4. $L_A>L_B$

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

Q. If torque acting on a system is zero, the quantity that remains constant is

1. force
2. linear momentum
3. angular momentum
4. angular velocity

Q. If a particle moves with a constant velocity, the angular momentum of this particle about any point

1. will vary linearly.
2. will remain constant.
3. will be always zero.
4. will vary quadratically.

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 man, standing on a turn-table, is rotating at a certain angular frequency with his arms outstretched. He suddenly folds his arms. If his moment of inertia with folded arms is $75\%$ of that with outstretched arms, his rotational kinetic energy will

1. Increase by $33.3\%$
2. Decrease by $33.3\%$
3. Increase by $25\%$
4. Decrease by $25\%$

Q. A particle $P$ is moving along a straight line as shown in the figure. During the motion of the particle from $A$ to $B$, the angular momentum of the particle about $\text{O}$

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

Q. A heavy solid sphere is thrown on a horizontal rough surface with initial velocity $u$ without rolling. What will be its speed in $\text{m/s}$ when it starts pure rolling? (Assume $u=7\text{m/s}$)

Q. One twirls a circular ring (of mass $M$ and radius $R$ ) near the tip of one's finger as shown in Figure $1.In$ the process the finger never loses contact with the inner rim of the ring. The finger traces out the surface of a cone, shown by the dotted line. The radius of the path traced out by the point where the ring and the finger is in contact is $r$. The finger rotates with an angular velocity ${\omega }_0$. The rotating ring rolls without slipping on the outside of a smaller circle described by the point where the ring and the finger is in contact (Figure 2 ). The cofficient of friction between the ring and the finger is $\mu$ and the acceleration due to gravity is $g$

1. $M{\omega }_0^2(R-r{)}^2$
2. $M{\omega }_0^2{R}^2$
3. $\frac{3}{2}M{o}_0^2(R-r{)}^2$
   
   
4. $\frac{1}{2}M{c}_0^2(R-r{)}^2$

Q. A 2 Kg mass is rotating on a circular path of radius 0.8 m with angular velocity of 44 rad/sec. If radius of the path becomes 1m, then what will be the value of angular velocity?

1. 8.12 rad/sec
2. 28.16 rad/sec

3. 35.26 rad/sec

4. 19.28 rad/sec

Q.

A bullet of mass 10 g and speed 500 m/s is fired into a door and gets embedded exactly at the centre of the door. The door is 1.0 m wide and weighs 12 kg. It is hinged at one end and rotates about a vertical axis practically without friction. Find the angular speed of the door just after the bullet embeds into it.
(Hint: The moment of inertia of the door about the vertical axis at one end is $M{L}^2/3$.)

Q. A solid disc of radius $20\text{cm}$ and mass $10\text{kg}$ is rotating with an angular velocity of $600\text{rpm}$, about an axis normal to its circular plane and passing through its centre of mass. The retarding torque required to bring the disc to rest in $10\text{s}$ is $\pi \times {10}^{-1}\text{Nm}$.

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

Q. A meter stick is pivoted about its centre. A piece of wax of mass $20\text{g}$ travelling horizontally and perpendicular to it at $10\text{m/s}$ strikes and adheres to one end of the stick so that the stick starts to rotate in a horizontal circle. Given the moment of inertia of the stick and wax about the pivot is $0.02{\text{kg-m}}^2$, the initial angular velocity of the stick is:

1. $2\text{rad/s}$
2. $3\text{rad/s}$
3. $4\text{rad/s}$
4. $5\text{rad/s}$

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 solid sphere of mass $M$ and radius $R$ is rolling without slipping as shown in figure. Find out the magnitude of total angular momentum of the sphere about point of contact $\left(P\right)$.

1. $L_{\text{Total}}=\frac{3}{5}M{V}_{0}R$
2. $L_{\text{Total}}=\frac{12}{5}M{V}_{0}R$
3. $L_{\text{Total}}=\frac{7}{5}M{V}_{0}R$
4. $L_{\text{Total}}=\frac{2}{5}M{V}_{0}R$

Q.

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

1. 

2. 

3. 

4. 

Q. 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 $\omega$ about the vertical axis passing through the center of $A$. The angular momentum of $B$ is $nM\omega R^2$ with respect to the center of $A$. Which of the following is the value of $n$?

1. $2$
2. $\frac{9}{2}$
3. $5$
4. $\frac{7}{2}$

Q. Moment of inertia of uniform rod of mass 'M' and length 'L' about an axis through its centre and perpendicular to its length is given by $\frac{M{L}^2}{12}$. Now consider one such rod pivoted at its centre, free to rotate in a vertical plane. The rod is at rest in the vertical position. A bullet of mass 'M'moving horizontally at a speed 'v' strikes and embedded in one end of the rod. The angular velocity of the rod just after the collision will be

1. 
2. 
3. 
4. 

Q. Consider a body, shown in figure, consisting of two identical balls, each of mass $2\text{kg}$ connected by a light rigid rod. If an impulse, $J=20\text{kg - m/s}$ is imparted to the body at one of its end, then angular velocity will be $\left(\text{in rad/s}\right)$

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

Q. A thin circular ring of mass 'M' and radius 'R' is rotating about its axis with a constant angular velocity $\omega$. Four objects each of mass 'm', are kept gently to the opposite ends of two perpendicular diameters of the ring. The new angular velocity of the ring will be

1. 
2. 
3. 
4. 

Q. A particle of mass $m$ is projected at time $t=0$ from a point $P$ on the ground with a speed $v_0$, at an angle of ${45}^{\circ }$ to the horizontal. What is the magnitude of the angular momentum of the particle about $P$ at time $t=\frac{v_0}{g}$?

1. $\frac{m{v}_0^2}{2\sqrt{2}g}$
2. $\frac{m{v}_0^2}{\sqrt{2}g}$
3. $\frac{m{v}_0^3}{2\sqrt{2}g}$
4. $\frac{m{v}_0^3}{\sqrt{2}g}$

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 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 rod of mass $m$ and length $l$ is lying along the y-axis such that its end is at the origin. Suddenly an impulse is given to the rod such that immediately after the impulse, the end on the origin has a velocity $v_0\hat{i}$ and the other end has a velocity $2v_o\hat{i}$. The magnitude of angular momentum of the rod about the origin at this instant is $\frac{5}{n}m{v}_{0}l$. Find n.

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

Q. Statement I: The velocity of centre of mass with respect to centre of mass frame is zero.
Statement II: Velocity of centre of mass with respect to ground's reference frame is always zero.

1. I is right; II is wrong
2. I is wrong; II is right
3. Both are wrong
4. Both are right