A rod of negligible mass and length $l$ is pivoted at the point $\frac{l}{4}$ distance from the left end as shown. A particle of mass $m$ is fixed to its left end and another particle of mass $2m$ is fixed to the right end. If the system is released from rest and after some time becomes vertical, the speed $v$ of the two masses and angular velocity at that instant is 

Two identical balls $A$ and $B$ of mass $m$ each is placed on a fixed wedge as shown in the figure. The ball $B$ is kept at rest and it is released just before two balls collide. The ball $A$ rolls down without slipping on an inclined plane and collides elastically with the ball $B$. The kinetic energy of the ball $A$ just after the collision with the ball $B$ is (Neglect friction between $A$ and $B$, also neglect the radius of the balls)

Statement-1: In an elastic collision between two bodies, the relative speed of the bodies after collision is equal to the relative speed before the collision.

Statement-2: In an elastic collision, the linear momentum of the system is conserved.

Look at the drawing given in the figure which has been drawn with ink of uniform line-thickness. The mass of ink used to draw each of the two inner circles, and each of the two line segments is$\mathrm{m}.$The mass of the ink used to draw the outer circle is $6\mathrm{m}.$ The coordinates of the centres of the different part are: outer circle $(0, 0),$ left inner circle$(–\mathrm{a}, \mathrm{a}),$right inner circle $(\mathrm{a}, \mathrm{a}),$ vertical line$(0, 0)$and horizontal line$(0, –\mathrm{a}).$The y-coordinate of the centre of mass of the ink in this drawing is

Two small particles of equal masses start moving in opposite directions from a point $A$ in a horizontal circular orbit. Their tangential velocities are $v \text{and} 2v,$respectively, as shown in the figure. Between collisions, the particles move with constant speeds. After making how many elastic collisions, other than that at $A,$ these two particles will again reach the point $A?$

A ball of mass$0.2 \mathrm{kg}$ rests on a vertical post of height $5 \mathrm{m}.$ A bullet of mass $0.01 \mathrm{kg},$ travelling with a velocity $V {\mathrm{ms}}^{-1}$ in a horizontal direction, hits the centre of the ball. After the collision, the ball and bullet travel independently. The ball hits the ground at a distance of $20 \mathrm{m}$and the bullet at a distance of $100 \mathrm{m}$ from the foot of the post. The initial velocity $V$ of the bullet is