The minimum horizontal force applied to the centre of wheel of radius $R$ and weight $w$ to pull it over a step of height $R/2$

 

A rod of mass $m$ and length $l$ hinged at its centre is placed on a horizontal surface. A bullet of mass $m$ moving with velocity $v$ strikes the end $A$ of the rod and gets embedded in it. The angular velocity with which the system rotates after the bullet strikes the rod

A uniform rod of length $6\mathrm{a}$ and mass $8\mathrm{m}$ lies on a smooth horizontal table. Two particles of masses $m$ and $2\mathrm{m}$ moving in the same horizontal plane with speed $2\mathrm{v}$ and $v$ respectively, strike the rod and stick to the rod after collision as shown in the figure. The velocity of centre of mass and angular velocity about centre of mass just after collision are respectively.

A ball of mass $\left(m\right) 0.5 \mathrm{kg}$ is attached to the end of a string having length $\left(L\right)$ $0.5 \mathrm{m}$. The ball is rotated on a horizontal circular path about the vertical axis. The maximum tension that the string can bear is $324 \mathrm{N}$. The maximum possible value of angular velocity of ball $(\text{in} \mathrm{radian} {\mathrm{s}}^{-1})$ is,

A particle of mass $m$ moving with velocity $v_0$ hits a stationary thin, uniform ring of equal mass $m$ laying horizontally on a smooth horizontal surface, tangentially and sticks to it. The angular speed of system after collision is

A uniform rod of mass $m$ and length $L$ with a particle of mass $m$ attached at one end is released from horizontal position. The other end of rod is hinged. The angular speed of rod when it becomes vertical

A smooth, uniform solid sphere, rolling without slipping with velocity $v_0$, collides elastically and head-on with a stationary identical sphere placed on a rough horizontal surface. The velocity of second sphere when it starts pure rolling, is

A reel of thread is released from rest by holding one end of thread at rest. Consider the reel as a hollow cylinder. The acceleration of centre of reel is