The Principle

Q. A cricket ball of mass 0.5 kg strikes a bat normally with a velocity of $30m{s}^{-1}$ and rebounds with a velocity of $20m{s}^{-1}$ in the opposite direction. The impulse of the force exerted by the ball on the bat is

1. 0.5 Ns
2. 1.0 Ns
3. 25 Ns
4. 50 Ns

Q.

8. A solid homogeneous sphere is moving on a rough horizontal surface, partly rolling and partly sliding During this kind of motion of the sphere:

(a) total kinetic energy is conserved

(b angular momentum of the sphere about the point of contact with the plane is conserved (c) only the rotational kinetic energy about centre of mass is conserved

(d) angular momentum about centre of mass is conserved

Q. a circular disc have mass 2kg and radius 10 cm rolls without slipping with speed 2 m/s the total kinetic energy of disc

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 ML²/3.)

Q. A ball of mass $m$ travelling at a speed $v$ hits a rod of the same mass $m$ and length $L$ as shown in the figure below. After the collision , the ball sticks to the rod.Find the speed of the ball after collision.

1. $\frac{4v}{5}$
2. $\frac{3v}{5}$
3. $\frac{2v}{5}$
4. $\frac{v}{5}$

Q. A thin circular ring of mass $M$ and radius $r$ is rotating about its axis with a constant angular velocity $\omega$. 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 of

1. $\frac{\omega M}{M+m}$
2. $\frac{\omega (M-2m)}{M+2m}$
3. $\frac{\omega M}{M+2m}$
4. $\frac{\omega (M+2m)}{M}$

Q. A uniform rod of length $l$ lies on a frictionless horizontal surface on which it is pivoted about its centre. A ball moving with speed $v$ as shown in figure collides with the rod at one of the ends. After hitting, the ball sticks to the rod. There is a similar uniform rod of length $l$, pivoted at one of its ends and a similar ball with same speed strikes the other end of the rod and sticks to it. The ratio of final angular momentum in both cases is:

1. $\frac{1}{2}$
2. $\frac{1}{3}$
3. $\frac{8}{1}$
4. $\frac{1}{4}$

Q. If external torque acting on the system along $x-$ axis add up to zero, then

1. Angular momentum of system along $x-$ axis is always zero.
2. Angular momentum of system along $y-$ axis remains constant.
3. Angular momentum of system along $z-$ axis remains constant
4. Angular momentum of system along $x-$ axis remain constant.

Q. A force $\overrightarrow{F}=\alpha \hat{i}+3\hat{j}+6\hat{k}$ is acting at a point $\overrightarrow{r}=2\hat{i}-6\hat{j}-12\hat{k}$. Find the value of $\alpha$ for which angular momentum about origin is conserved.

1. $\text{Zero}$
2. $1$
3. $-1$
4. $2$

Q. A turn table is rotatingin horizontalplane abouy its its own axis at an angular velocity 90rpm .while a person is on the turn table at its edge. If he gently walks to the center of table by which moment of inertia decreases by 25% than time period of rotationof turn table is

Q. Point masses $m_1$ and $m_2$ are placed at the opposite ends of a rigid rod of length L, and negligible mass. The rod is to be set rotating about an axis perpendicular to it. The position of point P on this rod through which the axis should pass so that the work required to set the rod rotating with angular velocity ${\omega }_0$ is minimum, is given by:

1. $x=\frac{m_{2}L}{m_1+m_2}$
2. $x=\frac{m_{1}L}{m_1+m_2}$
3. $x=\frac{m_1}{m_2}L$
4. $x=\frac{m_2}{m_1}L$

Q.

Suppose we have a box filled with gas and a piston is also attached at the top of the box. What are the ways of changing the state of gas (and hence its internal energy)? Answer could be more than one choice.

1. a) Move the box so that it has kinetic energy

2. b) Bring box in contact with a body with higher temperature

3. c) Pushing the piston down so as to do work on the system

4. d) a. and c. both.

Q. If external torque acting on the system along $x-$ axis add up to zero, then

1. Angular momentum of system along $x-$ axis is always zero.
2. Angular momentum of system along $y-$ axis remains constant.
3. Angular momentum of system along $z-$ axis remains constant
4. Angular momentum of system along $x-$ axis remain constant.

Q. what are the quantities that are same if the trajectory of two objects is the identical ?

Q. A man stands on a rotating platform, with his arms stretched horizontally holding a 5 kg weight in each hand. The angular speed of the platform is 30 revolutions per minute. The man then brings his arms close to his body with the distance of each weight from the axis changing from 90cm to 20cm. The moment of inertia of the man together with the platform may be taken to be constant and equal to 7.6 kg m². (a) What is his new angular speed? (Neglect friction.) (b) Is kinetic energy conserved in the process? If not, from where does the change come about?

Q. A small bead of mass $m$ moving with velocity $v$ gets threaded on a stationary semicircular ring of mass $m$ and radius $R$ kept on a horizontal table. The ring can freely rotate about its centre. The bead comes to rest relative to the ring. What will be the final angular velocity of the system?

1. $\frac{v}{R}$
2. $\frac{2v}{R}$
3. $\frac{v}{2R}$
4. $\frac{3v}{R}$

Q. A thin circular ring of mass $M$ and radius $R$ is rotating about its axis with a constant angular velocity $\omega$. 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

1. $\frac{\omega M}{m+M}$
2. $\frac{\omega (M-2m)}{M+2m}$
3. $\frac{\omega M}{M+2m}$
4. $\omega \left(\frac{M+2m}{M}\right)$

Q.

A composite rod of mass '2m' $\&$ length '2l' consists of two identical rods joined end to end at P. The composite rod is hinged at one of its ends and is kept horizontal. If it is released from rest, ·Find its angular speed when it becomes vertical

1. $\sqrt{\frac{3}{2}\frac{g}{l}},\sqrt{\frac{g}{2l}}$

2. $\sqrt{\frac{3g}{2l}},\sqrt{\frac{3g}{2l}}$

3. $\sqrt{\frac{3g}{2l}},\sqrt{\frac{g}{l}}$

4. $\sqrt{\frac{3g}{2l}},\sqrt{\frac{g}{3l}}$

Q. A smooth uniform rod of length $L$ and mass $M$ has two identical beads of negligible size, each of mass $m$, which can slide freely along the rod. Initially, the two beads are at the centre of the rod and the system is rotating with angular velocity ${\omega }_0$ about an axis perpendicular to the rod and passing through the mid-point $C$ of the rod. There are no external forces. When the beads reach the ends of the rod, then angular velocity of the system is

1. $\left(\frac{M}{M+3m}\right){\omega }_0$
2. $\left(\frac{M}{M+6m}\right){\omega }_0$
3. $\left(\frac{M+6m}{M}\right){\omega }_0$
4. ${\omega }_0$

Q. A thin uniform circular disc of mass $M$ and radius $R$ is rotating with an angular velocity $\omega ,$ in a horizontal plane about an axis passing through its centre and perpendicular to its plane. Another disc of same dimensions but of mass $\frac{M}{4}$ is placed gently on the first disc co-axially. The final angular velocity of the system is

1. $\frac{2}{3}\omega$
2. $\frac{4}{5}\omega$
3. $\frac{3}{4}\omega$
4. $\frac{1}{3}\omega$

Q. A disc of mass $m_0$ rotates freely about a fixed horizontal axis through its centre. A thin cotton pad is fixed to its rim, which can absorb water. The mass of water dripping onto the pad is $\mu$ per second. After what time will the angular velocity of the disc gets reduced to half of its initial value?

1. $\frac{2m_0}{\mu }$
2. $\frac{3m_0}{\mu }$
3. $\frac{m_0}{\mu }$
4. $\frac{m_0}{2\mu }$

Q. A thin uniform circular disc of mass $M$ and radius $R$ is rotating with an angular velocity $\omega ,$ in a horizontal plane about an axis passing through its centre and perpendicular to its plane. Another disc of same dimensions but of mass $\frac{M}{4}$ is placed gently on the first disc co-axially. The final angular velocity of the system is

1. $\frac{2}{3}\omega$
2. $\frac{4}{5}\omega$
3. $\frac{3}{4}\omega$
4. $\frac{1}{3}\omega$

Q. Find the radius of gyration of circular ring of radius r about a line perpendicular to the plane of the ring and passing through one of its particles.

Q. 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 with constant speed $v$ along a diameter of the disc to reach it's other end. During the journey of the insect, the angular speed of the disc will

1. Remains unchanged
2. Continously decreases
3. Continously increases
4. First increase and then decrease.

Q. In the given figure, a ball strikes a rod (hinged at point $A$) elastically. The rod is free to rotate in the vertical plane. Then, identify the correct statement.

1. Linear momentum of system of (ball + rod) is conserved.
2. Angular momentum of system of (ball+ rod) about hinged point $A$ is conserved.
3. Initial kinetic energy of the system is equal to final kinetic energy of the system.
4. Linear momentum of ball is conserved.

Q. 38. If L is the angular momentum of a staellite revolving around the earth in a circular orbit of radius r with speed v , then the relationship between L and v is ?

Q. 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

1. $\frac{K}{2}$
2. $\frac{K}{4}$
3. $2K$
4. $4K$

Q. A man of mass $100kg$ stands at the rim of a turntable of radius $2m$ and moment of inertia $4000kg{m}^2$ mounted on a vertical frictionless shaft at its center. The whole system is initially at rest. The man now walks along the outer edge if the turntable(anticlockwise) with a velocity of $1m/s$ relative to earth. With what angular velocity and in what direction does the turntable rotate?

1. The turntable rotates anticlockwise(in the direction of the man motion) with angular velocity $0.05rad/s$.
2. The turntable rotates clockwise(opposite to the man motion) with angular velocity $0.1rad/s$.
3. The turntable rotates clockwise(opposite to the man motion) with angular velocity $0.5rad/s$.
4. The turntable rotates anticlockwise(in the direction of the man motion) with angular velocity $0.5rad/s$.

Q.
A rod of mass 'M' and length 'L' is placed on frictionless horizontal surface. Rod is given an angular velocity ${\omega }_0$ as shown in figure. When rod rotates by ${90}^{\circ }$, it hits a stationary hinge and gets attached to it. New angular velocity is $\omega$ such that $\omega =n{\omega }_0$. Here n is . (Answer upto two digits after the decimal point.)

Q. A force $\overrightarrow{F}=\alpha \hat{i}+3\hat{j}+6\hat{k}$ is acting at a point $\overrightarrow{r}=2\hat{i}-6\hat{j}-12\hat{k}$. Find the value of $\alpha$ for which angular momentum about origin is conserved.

1. $\text{Zero}$
2. $1$
3. $-1$
4. $2$