An arrow sign is made by cutting and rejoining a quarter part of a square plate of side '$L$' as shown. The distance $OC$, where '$C$' is the centre of mass of the arrow, is

A block of mass $M$ is tied to one end of a massless rope. The other end of the rope is in the hands of a man of mass $2M$ as shown in the figure. The block and the man are resting on a rough wedge of mass $M$ as shown in the figure. The whole system is resting on a smooth horizontal surface. The man pulls the rope. Pulley is massless and frictionless. What is the displacement of the wedge when the block meets the pulley. (Man does not leve his position during the pull)

A bead can slide on a smooth straight wire and a particle of mass $\mathrm{m}$ attached to the bead by a light string of length $L$. The particle is held in contact with the wire and with the string taut and is then let fall. If the bead has mass $2\mathrm{m}$ then when the string makes an angle $\theta$ with the wire, the bead will have slipped a distance

A uniform metallic spherical shell is suspended from ceiling. It has two holes $A$ and $B$ at top and bottom respectively. Which of the following is/are true:-

If both the blocks as shown in the given arrangement are given together a horizontal velocity towards right. If $a_{\mathrm{cm}}$ be the subsequent acceleration of the centre of mass of the system of blocks then $a_{\mathrm{cm}}$ equals:-

Two persons $A$ and $B$ of weight $80 \mathrm{kg}$ and $50 \mathrm{kg}$ respectively are standing at opposite ends of a boat of mass $70 \mathrm{kg}$ and length $2 \mathrm{m}$ at rest. When they interchange their positions then displacement of the centre of mass of the boat will be:-