Torque of Internal Forces

Q. A mass $M$ attached to a horizontal spring, executes SHM with amplitude $A_1$ on a smooth horizontal surface. When the mass $M$ passes through its mean position, then a small mass $m$ is placed over it and both of them move together with amplitude $A_2$. The ratio of $\left(\frac{A_1}{A_2}\right)$ will be

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

Q. A string is wrapped around the rim of a wheel of moment of inertia $0.20kg-m^2$ and radius $20\text{cm}$. The wheel is free to rotate about its axis. Initially, the wheel is at rest. The string is now pulled by a force of $20\text{N}$. Find the angular velocity (rad/s) of the wheel after 5 seconds.

Q. A uniform bar of length $6a$ and mass $8m$ lies on a smooth horizontal table. Two point masses $m$ and $2m$, moving in the same horizontal plane with speeds $2v$ and $v$ respectively strike the bar (as shown in figure) and stick to the bar after collision. By denoting angular velocity (about centre of mass), total energy and velocity of centre of mass after collision as $\omega ,E$ and $v_o$ respectively, we can say that

1. $v_o=0$
2. $\omega =\frac{6v}{5a}$
3. $\omega =\frac{3v}{5a}$
4. $E=\frac{3}{5}m{v}^2$

Q. A ring of mass $m$ is rolling without slipping with linear speed $v$ as shown in figure. Four particles each of mass $m$ are also attached at points $\text{A, B, C}$ and $\text{D}$. Find total kinetic energy of the system.

1. $6m{v}^2$
   
2. $3m{v}^2$
   
3. $m{v}^2$
   
4. $5m{v}^2$
   

Q. A particle of mass $20\text{g}$ is released with an initial velocity $5{\text{ms}}^{-1}$ along the curve from the point $A$, as shown in the figure. The point $A$ is at height $h$ from point $B$. The particle slides along the frictionless surface. When the particle reaches point $B$, its angular momentum about $O$ will be:
$(\text{Take}g=10{\text{ms}}^{-2})$

1. $8{\text{kg-m}}^2{\text{s}}^{-1}$
2. $3{\text{kg-m}}^2{\text{s}}^{-1}$
3. $6{\text{kg-m}}^2{\text{s}}^{-1}$
4. $2{\text{kg-m}}^2{\text{s}}^{-1}$

Q. A mass of $5\text{kg}$ is moving along a circular path of radius 1 m. If the mass moves with 300 revolutions per minute, its kinetic energy would be

1. $250{\pi }^{2}J$
2. $100{\pi }^{2}J$
3. $5{\pi }^{2}J$
4. 0

Q. A uniform thin rod of length $l$ and mass $m$ is hinged at a distance $l/4$ from one of the end released from horizontal position as shown in figure. The angular velocity of the rod as it passes the vertical position is

1. $2\sqrt{\frac{5g}{7l}}$
2. $2\sqrt{\frac{6g}{7l}}$
3. $\sqrt{\frac{3g}{7l}}$
4. $2\sqrt{\frac{g}{l}}$

Q. A ring of mass ${}^{\prime }{m}^{\prime }$ is rolling without slipping with velocity of centre of mass $\left(v\right)$ as shown in figure. Four particles each of mass ${}^{\prime }{m}^{\prime }$ are also attached at points $A,B,C$ and $D$. Total kinetic energy of system is:

1. $5m{v}^2$
2. $3m{v}^2$
3. $2m{v}^2$
4. $8m{v}^2$

Q. Find the maximum angular displacement of the rod if a particle of mass $m$ moving with velocity $6\text{m/s}$ sticks to it after collision.

1. ${63}^{\circ }$
2. ${60}^{\circ }$
3. ${53}^{\circ }$
4. ${45}^{\circ }$

Q. A thin rod of mass $0.9\text{kg}$ and length $1\text{m}$ is suspended at rest from one end so that it can freely oscillates in the vertical plane. A particle of move $80\text{m/s}$ hits the rod at its bottom most point and stickes to it (see figure). The angular speed (in $\text{rad/s}$) of the rod immeditely after the collision will be____

Q. A running man has half the kinetic energy of a running boy of half his mass. The man speeds up by $1m/s$ and then has $K.E.$ as that of the boy. What were the original speeds of man and the boy?

1. $(\sqrt{2}+1)m/s;2(\sqrt{2}+1)m/s$
2. $(\sqrt{2}+1)m/s;\sqrt{2}m/s$
3. $\sqrt{2}m/s;(2\sqrt{2}-1)m/s$
4. $(\sqrt{2}-1)m/s;2(\sqrt{2}-1)m/s$

Q. In the figure shown below, if the block is a cube of side $1\text{m}$, then loss of mechanical energy during impact if the difference in vertical heights between the two horizontal surfaces is very small is:

1. $5\text{J}$
2. $6\text{J}$
3. $6\text{J}$
4. $8\text{J}$

Q. The reel shown in figure has radius $R$ and moment of inertia $I$. One end of the block of mass $m$ is connected to a spring of force constant $k$ and the other end is fastened to a cord wrapped around the reel. The reel's axle and incline are frictionless. The reel is wound counterclockwise so that the spring stretches a distance $d$ from its unstretched position and the reel is then released from rest. The angular speed of the reel when the spring is again unstretched is

1. $\sqrt{\frac{2mgd\sin \theta +k{d}^2}{I+m{R}^2}}$
2. $\sqrt{\frac{mgd\sin \theta +k{d}^2}{I+m{R}^2}}$
3. $\sqrt{\frac{2mgd\sin \theta -k{d}^2}{I+m{R}^2}}$
4. $\sqrt{\frac{mgd\sin \theta -k{d}^2}{I+m{R}^2}}$

Q. Consider a uniform rod $PQ$ of mass $M$ and length $L$ lying on a smooth horizontal surface. An impulse $J$ is applied to the end $Q$ in the horizontal direction as shown in the figure. The speed of a particle $R$ at a distance $\frac{L}{6}$ from the centre towards end $P$ of the rod after time $t=\frac{\pi ML}{12J}$ is

1. $\frac{\sqrt{2}J}{M}$
   
2. $\frac{2J}{M}$
   
3. $\frac{J}{\sqrt{2}M}$
   
4. $\frac{J}{M}$
   

Q. A uniform bar of length $6\text{m}$ and mass $16\text{kg}$ lies on a smooth horizontal table. Two point masses $2\text{kg}$ and $4\text{kg}$ moving in the square horizontal plane with $6\text{m/s}$ and $3\text{m/s}$ respectively strike the bar as shown in figure and stick to the bar after collision. Denote angular velocity (about the centre of mass), total energy and centre of mass velocity by $\omega ,E$ and $v_c$ respectively. Then after the collision:

1. $v_c=0\text{m/s}$
2. $\omega =\frac{9}{5}\text{rad/s}$
3. $\omega =\frac{3}{5}\text{rad/s}$
4. $E=\frac{54}{5}\text{J}$

Q. A bullet of mass $0.01kg$ is travelling at a speed of $500m/s$ strikes a block of mass $2kg$ which is suspended by a string of length $5m$. The centre of gravity of the block is found to rise a vertical distance of $0.2m$. What is the speed of the bullet after it emerges from the block?

Q. Two identical rods each of length $l$ and mass $m$ are welded together at right angle and then suspended from a knife-edge as shown. Angular frequency of small oscillations of the system in its own plane about the point of suspension is

1. $\sqrt{\frac{3g}{4\sqrt{2}l}}$
2. $\sqrt{\frac{3g}{2\sqrt{2}l}}$
3. $\sqrt{\frac{3g}{\sqrt{2}l}}$
4. None of these

Q. The net torque on a rigid body due to its internal forces is always zero.

1. True
2. False

Q. A string is wrapped around the rim of a wheel of moment of inertia $0.20kg-m^2$ and radius $20\text{cm}$. The wheel is free to rotate about its axis. Initially, the wheel is at rest. The string is now pulled by a force of $20\text{N}$. Find the angular velocity (rad/s) of the wheel after 5 seconds.

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. The net torque produced by the internal forces is ___

1. Positive
2. Negative
3. Zero
4. Cannot Say

Q. A square sheet of side 'a' is of mass 'M'.Its moment of inertia about a diagonal is:

1. $\frac{M{a}^2}{8}$
2. $\frac{M{a}^2}{6}$
3. $\frac{M{a}^2}{12}$
4. $\frac{M{a}^2}{9}$

Q. A body of mass $4m$ at rest explodes into three pieces. Two of the pieces each of mass $m$ move with a speed $v$ each in mutually perpendicular directions. The total energy released is:

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

Q. One mole of oxygen at ${27}^o$C and at one atmospheric pressure is enclosed in a vessel. Assuming the molecules to be moving with $V_{rms}$. Find the number of collisions per second which the molecules make with $1$ $m^2$ area of the vessel wall.

1. $6.4\times {10}^{27}$.
2. $5.4\times {10}^{27}$.
3. $4.4\times {10}^{27}$.
4. $1.97\times {10}^{27}$.

Q. A uniform wheel of mass $10.0kg$ and radius $0.400m$ is mounted rigidly on a massless axle through its center. The radius of the axle is $0.200m$, and the rotational inertia of the wheel-axle combination about its central axis is $0.600kg.m^2$.The wheel is initially at rest at the top of a surface that is inclined at angle $\theta ={30.0}^o$ with the horizontal; the axle rests on the surface while he wheel extends into a groove in the surface without touching the surface. Once released, the axle rolls down along he surface smoothly and without slipping. When the wheel-axle combination has moved down the surface by $2.00m$, what are its rotational kinetic energy?

Q. A uniform bar of length $6a$ and mass $8m$ lies on a smooth horizontal table. Two point masses $m$ and $2m$ moving in the same horizontal plane with speed $2v$ and $v$ respectively, strike the bar (as shown in figure) and stick to the bar after collision. C represents centre of mass of bar. Denoting angular velocity (about the centre of mass), total energy and centre of mass by $\omega$, E and $v_c$ respectively, then after collision we have

1. $v_c=0$
2. $\omega =\frac{3v}{5a}$
3. $\omega =\frac{v}{5a}$
4. $E=\frac{3m{v}^2}{5}$

Q. A dumb-bell consists of two identical small balls of mass $1/2kg$ each connected to the two ends of a $50cm$ long light rod. The dumb-bell is rotating about a fixed axis through the center of the rod and perpendicular to it at an angular speed of $10rad/s$. Am the impulsive force of average magnitude $5.0N$ acts on one of the masses in the direction of its velocity for $0.10s$. Find the new angular velocity of that system.

Q. A car of mass 10 metric ton rolls at $2m{s}^{-1}$ along a level track and collides with a loaded car of mass 20 metric ton, standing at rest. If the cars couple together, loss in K.E during collision is

1. $1.33J$
2. $1.33\times {10}^4$J
3. $1KJ$
4. $1.33KJ$

Q. A uniform bar of length $6a$ and mass $8m$ lies on a smooth horizontal table. Two point masses $m$ and $2m$, moving in the same horizontal plane with speeds $2v$ and $v$ respectively strike the bar (as shown in figure) and stick to the bar after collision. By denoting angular velocity (about centre of mass), total energy and velocity of centre of mass after collision as $\omega ,E$ and $v_o$ respectively, we can say that

1. $v_o=0$
2. $\omega =\frac{6v}{5a}$
3. $\omega =\frac{3v}{5a}$
4. $E=\frac{3}{5}m{v}^2$