$A$ particle executes uniform circular motion with angular momentum $L$. Its rotational kinetic energy becomes half,when the angular frequency is doubled. Its new angular momentum is

$A$ flywheel is an important part of a steam engine because it

$A$ thin uniform bar lies on a frictionless horizontal surface and is free to move in any way on the surface. Its mass is $0.300 \, kg$ and length is $2 \, m$. Two particles,each of mass $0.100 \, kg$,are moving on the same surface towards the two ends of the bar in a direction perpendicular to the bar. One particle moves with a velocity of $10 \, m/s$ towards one end,and the other moves with a velocity of $5 \, m/s$ towards the other end. If the collision between the particles and the bar is completely elastic and both particles strike the bar simultaneously,find the velocity of the centre of mass of the bar after the collision in $m/s$.

The total kinetic energy of a body of mass $10 \ kg$ and radius $0.5 \ m$ moving with a velocity of $2 \ m/s$ without slipping is $32.8 \ J$. The radius of gyration of the body is .......... $m$.

$ABC$ is an equilateral triangle with $O$ as its centre. $\vec F_1, \vec F_2$ and $\vec F_3$ represent three forces acting along the sides $AB, BC$ and $AC$ respectively. If the total torque about $O$ is zero,then the magnitude of $\vec F_3$ is

$A$ bullet of mass $10\, g$ and speed $500\, m/s$ is fired into a door and gets embedded exactly at the centre of the door. The door is $1.0\, m$ wide and weighs $12\, kg$. It is hinged at one end and rotates about a vertical axis practically without friction. The angular speed of the door just after the bullet embeds into it will be