Angular Impulse

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

Q.

A uniform chain of mass M and length L is held vertically in such a way that its lower end just touches the horizontal floor. The chain is released from rest in this position. Any portion that strikes the floor comes to rest. Assuming that the chain does not form a heap on the floor, calculate the force exerted by it on the floor when a length x has reached the floor.

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

Q. A rod of mass $2\text{kg}$ and length $1\text{m}$ is lying in the horizontal plane and pivoted about its one end. Initially, it is rotating about its pivoted end (axis perpendicular to horizontal plane) with an angular velocity $2\text{rad/s}$. Suddenly, an angular impulse $\overrightarrow{J}$ is given to the rod, because of which its angular velocity becomes $10\text{rad/s}$. Find the magnitude of angular impulse $\overrightarrow{J}$.

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

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

Q.

A cracker rocket is ejecting gases at a rate of 0.05kg/s with a velocity 400 m/s. What will be the accelerating force on the rocket?

Q. A pulley is rotated about its axis by a force $F=(20t-5t^2)\text{N}$ (where $t$ is measured in seconds) applied tangentially. Find the magnitude of angular impulse [in $\text{Nm-s}$] on the pulley in the initial $2\text{seconds}$. Given radius of pulley $=0.5\text{m}$.

1. $10/3$
2. $20/3$
3. $40/3$
4. zero

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

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 rigid rod of mass $m$ and length $L$ is shown in figure. A particle $P$ of mass $m$ moving with a speed $u$, normal to $AB$ strikes $A$ and sticks to it. Immediately after the impact, the velocity of $B$ w.r.t. $C$ is $n/10$, where $n$ is (in $\text{m/s}$)
[$C$ is the centre of mass of the system (rod+particle)]

Q. Find the angular impulse applied to the particle in the 4 sec.

1. $10{\text{kg-m}}^2/\text{sec}$
2. $40{\text{kg-m}}^2/\text{sec}$
3. $20{\text{kg-m}}^2/\text{sec}$
4. $5{\text{kg-m}}^2/\text{sec}$

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

1. 1
2. -1
3. 2
4. 0

Q. Find the angular impulse applied to the particle in the 4 sec.

1. $10{\text{kg-m}}^2/\text{sec}$
2. $40{\text{kg-m}}^2/\text{sec}$
3. $20{\text{kg-m}}^2/\text{sec}$
4. $5{\text{kg-m}}^2/\text{sec}$

Q. A particle of mass 2kg moves with an initial velocity u=(4i+4j)m/s.A constant force of F=-20j N is applied on the particle. Initially the particle was at (0, 0). The x-coordinate of the particle when its y coordinate again becomes zero is given by

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

Q. A solid sphere rests on a horizontal surface. A horizontal impulse is applied at height h from centre. The sphere starts rolling just after the application of impulse. The ratio $\frac{h}{R}$ will be :

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

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

Q. A uniform conductor of resistance R is cut into $10$ equal pieces. Half of them are joined in series and the remaining half of them are connected in parallel. If these two combination are joined in series, then the effective resistance of the pieces is?

1. $\frac{R}{2}$
2. $\frac{13}{25}$R
3. $\frac{R}{15}$
4. $\frac{25}{11}$R

Q. A object of mass $M$ and radius $R$ is performing pure rolling motion on a smooth horizontal surface under the action of a constant force $F$ as shown in figure. The object may be

1. ring
2. disk
3. hollow sphere
4. solid cylinder

Q. The bob in a simple pendulum of length $\ell$ is released at $t=0$ from the position of small angular displacement $\theta$. Linear displacement of the bob at any time $t$ from the mean position is given by

1. $\ell \theta \cos \sqrt{\frac{g}{l}}t$
2. $\ell \sqrt{\frac{g}{l}}t\cos \theta$
3. $\ell g\sin \theta$
4. $\ell \theta \sin \sqrt{\frac{g}{l}}t$

Q. The angular momentum about the center of mass shortly before and after the collision

1. Increases
2. Decreases
3. Remains same
4. None of these

Q. A thing rod of mass M = 6 kg and length L = 1/4 m is hanging vertically from the pivot P as shown in figure. A small block of mass m = 2 kg travelling horizontally with speed $V^o$ strikes at the bottom of the rod and sticks to it. Find the initial velocity (in m/s) with which block should move so that the rod becomes momentarily horizontal.

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 ring of mass 2kg carrying a current 4A is placed on a smooth horizontal surface as shown in the figure. Now a uniform magnetic field of 10T is switched on in the plane of the ring horizontally. The initial angular acceleration of the ring will be (in $rad/s^2)$ :

1. $40\pi$
2. $80\pi$
3. None of these
4. $20\pi$

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

Q. Two electric heater wires are made from the same material and have the same dimensions. First these are connected in series and then these are connected in parallel (to a 220 V AC source). Then the ratio of heat generated in the first case to the second case will be:

1. 1 : 2
2. 4 : 1
3. 1 : 4
4. 2 : 1

Q. A uniform plank of mass $m$ free to move in horizontal direction only is placed on the top a cylinder of mass $2m$ and radius $R$.The plank is attached to a fixed wall by means of a light spring of spring constant $K$. There is no slipping between the cylinder and the plank and between cylinder and ground. The time period of small oscillations of system is

1. $2\pi \sqrt{\frac{7m}{4k}}$
2. $\sqrt{\frac{k}{3m}}$
3. $2\pi \sqrt{\frac{3m}{k}}$
4. $\frac{1}{2\pi }\sqrt{\frac{k}{3m}}$