$A$ thin uniform bar of length $L$ and mass $8m$ lies on a smooth horizontal table. Two point masses $m$ and $2m$ move in the same horizontal plane from opposite sides of the bar with speeds $2v$ and $v$ respectively. The masses stick to the bar after collision at distances $L/3$ and $L/6$ respectively from the center of the bar. If the bar starts rotating about its center of mass as a result of the collision,the angular speed of the bar will be

$A$ rod of mass $m$ and length $L$,pivoted at one of its ends,is hanging vertically. $A$ bullet of the same mass moving at speed $v$ strikes the rod horizontally at a distance $x$ from its pivoted end and gets embedded in it. The combined system now rotates with angular speed $\omega$ about the pivot. The maximum angular speed $\omega_M$ is achieved for $x=x_M$. Then
$(A)$ $\omega=\frac{3 v x}{ L ^2+3 x^2}$
$(B)$ $\omega=\frac{12 v x}{L^2+12 x^2}$
$(C)$ $x_M=\frac{L}{\sqrt{3}}$
$(D)$ $\omega_M=\frac{v}{2 L} \sqrt{3}$

An object has a moment of inertia of $3 \ kg \cdot m^2$. It is rotating with an angular velocity of $2 \ rad/s$. If a mass of $12 \ kg$ is moving with a velocity of $v \ m/s$,at what value of $v$ will their kinetic energies be equal?

Consider a body of mass $1.0 \ kg$ at rest at the origin at time $t=0$. $A$ force $\overrightarrow{F}=(\alpha t \hat{i}+\beta \hat{j})$ is applied on the body,where $\alpha=1.0 \ Ns^{-1}$ and $\beta=1.0 \ N$. The torque acting on the body about the origin at time $t=1.0 \ s$ is $\vec{\tau}$. Which of the following statements is (are) true?
$(A)$ $|\vec{\tau}|=\frac{1}{3} \ Nm$
$(B)$ The torque $\vec{\tau}$ is in the direction of the unit vector $+\hat{k}$
$(C)$ The velocity of the body at $t=1 \ s$ is $\overrightarrow{v}=\frac{1}{2}(\hat{i}+2 \hat{j}) \ ms^{-1}$
$(D)$ The magnitude of displacement of the body at $t=1 \ s$ is $\frac{1}{6} \ m$

$A$ uniform metallic rod rotates about its perpendicular bisector with constant angular speed. If it is heated uniformly to raise its temperature slightly,what happens to its speed of rotation?

$A$ circular hoop 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 hoop and gets embedded in it. The thickness of the hoop is much smaller than $R$. The angular velocity with which the system rotates after the bullet strikes the hoop is