A child is standing at one end of a long trolley moving with a speed  $v$  on a smooth horizontal floor. If the child starts running towards the other end of the trolley with a speed $u,$  the centre of mass of the system (trolley + child) will move with a speed.

A wheel of radius $r$ rolls without slipping with a speed $v$ on a horizontal road. When it is at a point $A$ on the road, a small jump of mud separates from the wheel at its highest point $B$ and drops at point $C$ on the road. The distance $AC$ will be

The instantaneous angular position of a point on a rotating wheel is given by the equation $\theta (t) = 2t^3 -6t^2$. The torque on the wheel becomes zero at $t$ $=$ ........ $\sec.$

Figure shows a thin metallic triangular sheet $ABC.$ The mass of the sheet is $M.$ The moment of inertia of the sheet about side $AC$ is

In the following figure, a body of mass $m$ is tied at one end of a light string and this string and this string is wrapped around the solid cylinder of mass $M$ and radius $R$. At the moment $t = 0$ the system starts moving. If the friction is negligible, angular velocity at time $t$ would be

A disc is performing pure rolling on a smooth stationary surface with constant angular velocity as shown in figure. At any instant, for the lower most point of the disc -