The Principle

Q. Match the following and choose the correct option

Column I(initially)

Column II(when rolling withoutsliping begins)

i.		a	$V_{CM}$ is towards left in case of uniform ring
ii.		b	$V_{CM}$ is towards left in case of solid uniform sphere
iii.		c	$V_{CM}$ is towards right in case of uniform ring
iv.		d	$V_{CM}$ is towards right in case of solid uniform sphere

1. i$\to$a, b, ii$\to$b, c, iii$\to$a, b, iv$\to$a, b
2. i$\to$a, d, ii$\to$b, c, iii$\to$a, b, iv$\to$a, b
3. i$\to$a, b, ii$\to$c, d, iii$\to$a, d, iv$\to$a, b
4. i$\to$a, d, ii$\to$b, c, iii$\to$a, d, iv$\to$a, d

Q. A dancer is rotating on a smooth horizontal floor with an angular momentum $L$. The dancer folds her hands so that her moment of inertia decreases by $25\%$. The new angular momentum is

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

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 man of mass $M=50\text{kg}$ standing on the edge of a platform (moment of inertia of the platform is $I$ and radius $R=1\text{m}$) is rotating in anticlockwise direction at an angular speed of $20\text{rad/s}$. The man starts walking along the rim with a speed $1\text{m/s}$ relative to the platform, also in the anticlockwise direction. The new angular speed of the platform in $\left(\text{rad/s}\right)$ is (take $(I=0.02{\text{kg-m}}^2)$)

Q. Two discs are rotating about their axes, normal to the plane of the discs and passing through the centre of the discs. Disc $D$ has $2\text{kg}$ mass and $0.2\text{m}$ radius and initial angular velocity of $50\text{rad/s}$. Disc $D_2$ has $4\text{kg}$ mass, $0.1\text{m}$ radius and initial angular velocity of $200\text{rad/s}$. The two discs are brought in contact face to face with their axes of rotation coincident. The final angular velocity (in $\text{rad/s}$) of the system is

1. $50\text{rad/s}$
2. $20\text{rad/s}$
3. $100\text{rad/s}$
4. $200\text{rad/s}$

Q. A horizontal disc rotating freely about a vertical axis through its centre makes $90$ revolutions per minute. A small piece of wax of mass $m$ falls vertically on the disc and sticks to it at a distance $r$ from the axis. If the number of revolutions per minute reduces to $60$, the moment of inertia of the disc is

1. $m{r}^2$
2. $\frac{3}{2}m{r}^2$
3. $2m{r}^2$
4. $3m{r}^2$

Q. Two circular discs of moment of inertia $0.4{\text{kg-m}}^2$ and $0.6{\text{kg-m}}^2$ are rotating about their own axis (vertical axis passing through geometric centre) at the rate of $200\text{rpm}$ and $50\text{rpm}$ both in clockwise direction. If both are gently brought in contact to rotate about same axis, the angular speed of the combination is

1. $11.5\text{rad/sec}$
2. $23\text{rad/sec}$
3. $0\text{rad/sec}$
4. $5.5\text{rad/sec}$

Q. A point object of mass $m$ moving horizontally hits the uniform solid disc and stick to it. The disc is resting on a horizontal frictionless surface and pivoted at its centre as shown in figure. Find out the $MOI$ of the disc if it rotates with angular velocity $\omega =\frac{v}{5R}$ after the strike.

1. $I=\frac{m{R}^2}{2}$
2. $I=m{R}^2$
3. $I=4m{R}^2$
4. $I=\frac{3}{2}m{R}^2$

Q. A sphere is released from the top of a smooth inclined plane . When it moves downwards, its angular momentum will be

1. conserved about any point
2. conserved about the point of contact only
3. conserved about the centre of the sphere only
4. conserved about any point on a fixed line parallel to the inclined plane and passing through the centre of the sphere

Q. A horizontal disc rotating freely about a vertical axis makes $200\text{rpm}$. A piece of wax of mass $20\text{g}$ falls verticaly on the disc and adheres to it at a distance of $8\text{cm}$ from the axis. If the number of revolutions per minute is thereby reduced to $120\text{rpm}$, then calculate the moment of inertia of disc.

1. $3.8\times {10}^{-4}{\text{kg-m}}^2$
2. $1.92\times {10}^{-4}{\text{kg-m}}^2$
3. $1.28\times {10}^{-4}{\text{kg-m}}^2$
4. $2.56\times {10}^{-4}{\text{kg-m}}^2$

Q. A conical pendulum as shown in figure is rotating at angular velocity '$\omega$' about suspension point P, then for pendulum

1. angular momentum about P is conserved.
2. torque due to tension about P is zero.
3. angular momentum about axis of rotation is conserved.
4. net torque about P is zero.

Q. A very small stone is tied to a string of length 1m. The string is clamped at A. A tangential force of 10 N is applied on the stone. What will be the rate of change of angular momentum of the stone with respect to point A?

1. 20 N-m
2. 15 N-m
3. 30 N-m
4. None of these

Q. A point object of mass $m=1\text{kg}$ moving horizontally with $10\text{m/s}$ hits the lower end of the uniform thin rod of length $2\text{m}$ and sticks to it. The rod is rested on a horizontal, frictionless surface and pivoted at the other end as shown in figure. Find out the angular velocity of the system just after collision.

1. $15\text{rad/s}$
2. $3.75\text{rad/s}$
3. $7.5\text{rad/s}$
4. Zero

Q. A disc of mass $4m$ is resting on a horizontal frictionless surface and pivoted at point $\text{O}$ as shown in the figure, perpendicular to its plane. A point object of mass $m$, moving horizontally with velocity $v$, hits the disc and sticks to it. Find the angular velocity of the system after the hit.

1. $\frac{3v}{22R}$
2. $\frac{v}{11R}$
3. $\frac{2v}{R}$
4. $\frac{v}{6R}$

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 an angular velocity ${\omega }_0$ about an axis perpendicular to the rod and passing through the mid-point of the rod (see figure). There are no external forces. When the beads reach the ends of the rod, the angular velocity of the system is

1. $\omega =\frac{m{\omega }_0}{m+6M}$
2. $\omega =\frac{M{\omega }_0}{6M+m}$
3. $\omega =\frac{6m{\omega }_0}{M+6m}$
4. $\omega =\frac{M{\omega }_0}{M+6m}$

Q. The angular speed of a body changes from ${\omega }_1$ to ${\omega }_2$ due to changes in its moment of inertia without applying torque. The ratio of radius of gyration in the two cases is

1. ${\omega }_1:{\omega }_2$
2. $\sqrt{{\omega }_2}:\sqrt{{\omega }_1}$
3. $\sqrt{{\omega }_2^2}:\sqrt{{\omega }_1^2}$
4. $\sqrt{{\omega }_2^3}:\sqrt{{\omega }_1^3}$

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. A boy of mass $M$ stands at the edge of a circular platform of radius $R$ capable of rotating freely about its axis. The moment of inertia of the platform about the axis is $I$. The system is at rest. A friend of the boy throws a ball of mass $m$ with a velocity $v$ horizontally. The boy on the platform catches it when it passes tangentially to the platform. Find the angular velocity of the system after the boy catches the ball.

1. $\frac{mvR}{I+m{R}^2}$
2. $\frac{mvR}{1+M{R}^2}$
3. $\frac{mVR}{(M+m)R^2}$
4. $\frac{mvR}{1+(M+m)R^2}$

Q. A dancer is rotating on a smooth horizontal floor with an angular momentum $L$. The dancer folds her hands so that her moment of inertia decreases by $25\%$. The new angular momentum is

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

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 $5\text{m/s}$ strikes and adheres to one end of the stick which 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 angular velocity of the stick is

1. $1.58\text{rad/s}$
2. $2.24\text{rad/s}$
3. $2.50\text{rad/s}$
4. $5.00\text{rad/s}$

Q. A uniform rod of mass $m$ and length $l$ is free to rotate about a fixed horizontal axis passing through hinge. A small ball of mass $m$ moving horizontally with velocity $v_o$ strikes the rod at depth $y$ from hinge as shown in figure. If angular velocity of rod and velocity of ball just after the collision, be $\omega$ and $v$ respectively, then which of the following statements are wrong?

1. If $y=\frac{l}{2}$, then in case of elastic collision, $v=\frac{v_o}{2}$ and $\omega =\frac{v_o}{l}$
2. If $y=\frac{l}{2}$, then in case of elastic collision, momentum of ball-rod system remains conserved
3. If momentum of ball-rod system remains conserved then $y<\frac{l}{2}$
4. If $y=\frac{l}{2}$, then in case of elastic collision, magnitude of $v=\frac{v_o}{7}$ and $\omega =\frac{12v_o}{7l}$

Q. A disc of mass $4m$ is resting on a horizontal frictionless surface and pivoted at point $\text{O}$ as shown in the figure, perpendicular to its plane. A point object of mass $m$, moving horizontally with velocity $v$, hits the disc and sticks to it. Find the angular velocity of the system after the hit.

1. $\frac{3v}{22R}$
2. $\frac{v}{11R}$
3. $\frac{2v}{R}$
4. $\frac{v}{6R}$

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 along a diameter of the disc to reach its other end. During the journey fo the insect, the angular speed of the disc.

1. remains unchanged
2. continuously decreases
3. continuously increases
4. first increases and then decreases

Q. A disc of mass $m$ and radius $R$ is free to rotate in a horizontal plane about a vertical smooth fixed axis passing through its centre. There is a smooth groove along the diameter of the disc and two small balls of mass $m/2$ each, are placed in it on either side of the centre of the disc as shown in the figure. The disc is given an initial angular velocity ${\omega }_0$ and released.

The angular speed of the disc when the balls reach the end of the disc is

1. $\frac{{\omega }_0}{2}$
2. $\frac{{\omega }_0}{3}$
3. $\frac{2{\omega }_0}{3}$
4. $\frac{{\omega }_0}{4}$

Q. A man of mass $M=50\text{kg}$ standing on the edge of a platform (moment of inertia of the platform is $I$ and radius $R=1\text{m}$) is rotating in anticlockwise direction at an angular speed of $20\text{rad/s}$. The man starts walking along the rim with a speed $1\text{m/s}$ relative to the platform, also in the anticlockwise direction. The new angular speed of the platform in $\left(\text{rad/s}\right)$ is (take $(I=0.02{\text{kg-m}}^2)$)

Q. A uniform rod of mass $m$ and length $l$ is free to rotate about a fixed horizontal axis passing through hinge. A small ball of mass $m$ moving horizontally with velocity $v_o$ strikes the rod at depth $y$ from hinge as shown in figure. If angular velocity of rod and velocity of ball just after the collision, be $\omega$ and $v$ respectively, then which of the following statements are wrong?

1. If $y=\frac{l}{2}$, then in case of elastic collision, $v=\frac{v_o}{2}$ and $\omega =\frac{v_o}{l}$
2. If $y=\frac{l}{2}$, then in case of elastic collision, momentum of ball-rod system remains conserved
3. If momentum of ball-rod system remains conserved then $y<\frac{l}{2}$
4. If $y=\frac{l}{2}$, then in case of elastic collision, magnitude of $v=\frac{v_o}{7}$ and $\omega =\frac{12v_o}{7l}$

Q. A turntable rotates about a fixed vertical axis, making one revolution in $10\text{s}$. The moment of inertia of the turntable about the axis of rotation is $1200{\text{kg.m}}^2$. A man of mass $80\text{kg}$, initially standing at the centre of the turntable, runs outwards along the radius. What is the approximate angular velocity of the turntable when the man is at a distance of $2\text{m}$ from the centre?

1. $1.25\text{rad/s}$
2. $2.75\text{rad/s}$
3. $0.50\text{rad/s}$
4. $0.1\text{rad/s}$
   

Q. A cubical block of side $a$ is 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

1. $\frac{3v}{4a}$
2. $\frac{3v}{2a}$
3. $\sqrt{\frac{3v}{2a}}$
4. $\text{Zero}$

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. A disc of mass $M$ radius $R$ is fixed about an axis as shown in figure. If a particle of mass $\frac{M}{4}$ moving with speed $v$ perpendicular to the disc plane, hits the disc at point $P$ and sticks to the disc, the disc starts rotating about its axis as shown. Find the angular velocity of the system (disc + particle) after the hit.

1. $\omega =\frac{v}{R}$
2. $\omega =\frac{v}{5R}$
3. $\omega =\frac{v}{2R}$
4. $\omega =\frac{v}{\sqrt{5}R}$