Angular Impulse

Q. A uniform solid sphere is placed on a smooth horizontal surface. An impulse $I$ is given horizontally to the sphere at a height $h=\frac{4R}{5}$ above the centre line. ${}^{\prime }{m}^{\prime }$ and ${}^{\prime }{R}^{\prime }$ are the mass and radius of the sphere respectively.


Find angular velocity $\left(\omega \right)$ of the sphere and linear velocity $\left(v\right)$ of the centre of mass of the sphere after the impulse.

1. $\omega =\frac{I}{mR},v=\frac{2I}{m}$
2. $\omega =\frac{2I}{mR},v=\frac{I}{m}$
3. $\omega =\frac{I}{mR},v=\frac{I}{m}$
4. $\omega =\frac{2I}{mR},v=\frac{2I}{m}$

Q.

Figure shows two cylinders of radii $r_{1}and{r}_2$ having moments of inertia $I_{1}and{I}_2$ about their respective axes. Initially, the cylinders rotate about their axes with angular speeds ${\omega }_{1}and{\omega }_2$ as shown in the figure. The cylinders are moved closer to touch each other keeping the axes parallel. The cylinders first slip over each other at the contact but the slipping finally ceases due to the friction between them. Find the angular speeds of the cylinder 1 and 2 respectively after the slipping ceases.

1. $\left[\frac{I_1{\omega }_1{r}_2+I_2{\omega }_2{r}_1}{I_2{r}_1^2+I_1{r}_2^2}\right]r_2,\left[\frac{I_1{\omega }_1{r}_2+I_2{\omega }_2{r}_1}{I_2{r}_2^2+I_1{r}_1^2}\right]r_2$

2. $\left[\frac{I_1{\omega }_1{r}_2+I_2{\omega }_2{r}_1}{I_2{r}_1^2+I_1{r}_2^2}\right]r_2,\left[\frac{I_1{\omega }_1{r}_2+I_2{\omega }_2{r}_1}{I_2{r}_1^2+I_1{r}_2^2}\right]r_1$

3. $\left[\frac{I_1{\omega }_1{r}_2+I_2{\omega }_2{r}_1}{I_1{r}_1^2+I_2{r}_2^2}\right]r_2,\left[\frac{I_1{\omega }_1{r}_2+I_2{\omega }_2{r}_1}{I_1{r}_1^2+I_2{r}_2^2}\right]r_1$

4. $\left[\frac{I_1{\omega }_1{r}_2+I_2{\omega }_2{r}_1}{I_2{r}_1^2+I_1{r}_2^2}\right]r_2,\left[\frac{I_1{\omega }_1{r}_2+I_2{\omega }_2{r}_1}{I_2{r}_2^2+I_1{r}_1^2}\right]r_1$

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. Through what angle will it have rotated when the man reaches his initial position on the turntable?

1. The table rotates through $\frac{2\pi }{11}$ radians clockwise
2. The table rotates through $\frac{4\pi }{11}$ radians clockwise
3. The table rotates through $\frac{4\pi }{11}$ radians anticlockwise
4. The table rotates through $\frac{2\pi }{11}$ radians anticlockwise

Q. A uniform disc of mass $m$ and radius $R$ is placed on a smooth horizontal surface. An impulse $I$ is given horizontally to the sphere at height $h=\frac{3R}{5}$ above the centre line. Then which of the following option(s) is correct?

1. The minimum time after which the highest point touches the ground is $\frac{\pi Rm}{2I}$
2. The minimum time after which the highest point touches the ground is $\frac{5\pi Rm}{6I}$
3. The displacement of the COM during this interval is $\frac{5\pi R}{6}$
4. The displacement of the COM during this interval is $\frac{\pi R}{2}$

Q. A uniform rod of mass $m$, length $l$ is placed at rest on a smooth horizontal surface along $y-$ axis as shown in figure. An impulsive force $F$ is applied for a very small time interval $\Delta t$ along $x$ direction at point $A$. What will be the $x-$ coordinate of the end $A$ of the rod when the rod becomes parallel to $x-$ axis for the first time? Take origin at the mid point of the rod.

1. $\frac{\pi l}{12}$
2. $\frac{l}{2}(1+\frac{\pi }{12})$
3. $\frac{l}{2}(1-\frac{\pi }{6})$
4. $\frac{l}{2}(1+\frac{\pi }{6})$

Q. A rod of mass ${}^{\prime }{M}^{\prime }$ and length ${}^{\prime }{L}^{\prime }$ lies on horizontal table and hinged at its one end. A particle of mass ${}^{\prime }{m}^{\prime }$ is moving with velocity ${}^{\prime }{v}^{\prime }$ and it hits perpendicularly to other end of the rod. If particles sticks to the rod after collision, then find the angular velocity of the rod.

1. $\frac{2mv}{(M+3m)L}$
2. $\frac{3mv}{(M+3m)L}$
3. $\frac{2mv}{(M+2m)L}$
4. $\frac{3mv}{(M+2m)L}$

Q. A uniform rod of mass $m$, length $l$ is placed at rest on a smooth horizontal surface along $y-$ axis as shown in figure. An impulsive force $F$ is applied for a very small time interval $\Delta t$ along $x$ direction at point $A$. What will be the $x-$ coordinate of the end $A$ of the rod when the rod becomes parallel to $x-$ axis for the first time? Take origin at the mid point of the rod.

1. $\frac{\pi l}{12}$
2. $\frac{l}{2}(1+\frac{\pi }{12})$
3. $\frac{l}{2}(1-\frac{\pi }{6})$
4. $\frac{l}{2}(1+\frac{\pi }{6})$

Q. A uniform disk (m = 2 kg, r = 50 cm) is rotating at an angular speed 480 rpm. A second disk (m = 1 kg, r = 160 cm) is rotating at speed 360 rpm. Now first disk is dropped gently on second disk & they eventually rotate together about common axis. Find final angular velocity in rad/s.

1. 12.6 $\pi$
2. 11.2$\pi$
3. 22.4 $\pi$
4. 6.4 $\pi$

Q. A system consists of two identical small balls of mass $2\text{kg}$ each connected to the two ends of a $1\text{m}$ long light rod. The system is rotating about a fixed axis through the centre of the rod and perpendicular to it at an angular speed of $9\text{rad/s}$. An impulsive force of average magnitude $10\text{N}$ acts on one of the masses in the direction of its velocity for $0.20\text{s}$. Calculate the new angular velocity of the system.

1. $9\text{rad/s}$
2. $8\text{rad/s}$
3. $10\text{rad/s}$
4. $5\text{rad/s}$

Q.

Figure shows two cylinders of radii $r_{1}and{r}_2$ having moments of inertia $I_{1}and{I}_2$ about their respective axes. Initially, the cylinders rotate about their axes with angular speeds ${\omega }_{1}and{\omega }_2$ as shown in the figure. The cylinders are moved closer to touch each other keeping the axes parallel. The cylinders first slip over each other at the contact but the slipping finally ceases due to the friction between them. Find the angular speeds of the cylinder 1 and 2 respectively after the slipping ceases.

1. $\left[\frac{I_1{\omega }_1{r}_2+I_2{\omega }_2{r}_1}{I_2{r}_1^2+I_1{r}_2^2}\right]r_2,\left[\frac{I_1{\omega }_1{r}_2+I_2{\omega }_2{r}_1}{I_2{r}_2^2+I_1{r}_1^2}\right]r_2$

2. $\left[\frac{I_1{\omega }_1{r}_2+I_2{\omega }_2{r}_1}{I_2{r}_1^2+I_1{r}_2^2}\right]r_2,\left[\frac{I_1{\omega }_1{r}_2+I_2{\omega }_2{r}_1}{I_2{r}_1^2+I_1{r}_2^2}\right]r_1$

3. $\left[\frac{I_1{\omega }_1{r}_2+I_2{\omega }_2{r}_1}{I_1{r}_1^2+I_2{r}_2^2}\right]r_2,\left[\frac{I_1{\omega }_1{r}_2+I_2{\omega }_2{r}_1}{I_1{r}_1^2+I_2{r}_2^2}\right]r_1$

4. $\left[\frac{I_1{\omega }_1{r}_2+I_2{\omega }_2{r}_1}{I_2{r}_1^2+I_1{r}_2^2}\right]r_2,\left[\frac{I_1{\omega }_1{r}_2+I_2{\omega }_2{r}_1}{I_2{r}_2^2+I_1{r}_1^2}\right]r_1$

Q. A system consists of two identical small balls of mass $2\text{kg}$ each connected to the two ends of a $1\text{m}$ long light rod. The system is rotating about a fixed axis through the centre of the rod and perpendicular to it at an angular speed of $9\text{rad/s}$. An impulsive force of average magnitude $10\text{N}$ acts on one of the masses in the direction of its velocity for $0.20\text{s}$. Calculate the new angular velocity of the system.

1. $9\text{rad/s}$
2. $8\text{rad/s}$
3. $10\text{rad/s}$
4. $5\text{rad/s}$

Q. A circular wooden loop of mass $m$ and radius $R$ rests flat on a horizontal frictionless surface. A bullet, also of mass $m$, and moving with a velocity $V$, strikes the loop at the bottommost point and gets embedded in it. The thickness of the loop is much smaller than $R$. The angular velocity with which the system rotates just after the bullet strikes the loop is

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