A man weighing  $80 \mathrm{kg}$ is standing in a trolley weighing  $320 \mathrm{kg}$. The trolley is resting on frictionless horizontal rails. If the man starts walking on the trolley with a speed of $1 \mathrm{m} {\mathrm{s}}^{-1}$, then after $4 \mathrm{s}$, his displacement relative to the ground will be

A small ring is rolling without slipping on the circumference of a large bowl as shown below in the figure. The ring is moving down at $P_1$, comes down to the lower most point $P_2$ and is climbing up at $P_3$. Let ${\overrightarrow{v}}_{CM}$ denote the velocity of the centre of mass of the ring. Choose the correct statement regarding the frictional force on the ring.

If the sum of external forces acting on a system is zero, then the centre of mass

In figure $E$ and $V_{\mathrm{cm}}$ represent the total energy and speed of centre of mass of an object of mass $1 \mathrm{kg}$ in pure rolling. The object is:

A plank is moving in a horizontal direction with a constant acceleration $a\hat{\mathrm{i}}$. A uniform rough cubical block of side $l$ rests on the plank and is at rest relative to the plank.

Let the centre of mass of the block be at $\left(0, \frac{l}{2}\right)$ at a given instant. If $a=\frac{g}{10}$, then the normal reaction exerted by the plank on the block at that instant acts at

Four particles $\mathrm{A},\mathrm{ }\mathrm{B},\mathrm{ }\mathrm{C}$ and $\mathrm{D}$ with masses ${\mathrm{m}}_{\mathrm{A}}=\mathrm{m},{\mathrm{m}}_{\mathrm{B}}=2\mathrm{m},{\mathrm{m}}_{\mathrm{C}}=3\mathrm{m}$  and ${\mathrm{m}}_{\mathrm{D}}=4\mathrm{m}$ are at the corners of a square. They have accelerations of equal magnitude with directions as shown. The acceleration of the centre of mass of the particles is:

Two masses $A$ and $B$, each of mass $M$ are fixed together by a massless spring, A force acts on the mass $B$ as shown in figure. If the mass $A$ starts moving away from mass $B$ with acceleration $a$, then the acceleration of mass $B$ will be :

An object of mass $10 \mathrm{kg}$ is launched from the ground at $t=0,$ at an angle of $37°$ above the horizontal with a speed of $30 \mathrm{m}/\mathrm{s}.$ At some time after its launch, an explosion splits the projectile into two pieces. One piece of mass $4 \mathrm{kg}$ is observed at $(105 \mathrm{m}, 43 \mathrm{m})$ at $t=2 \mathrm{s}.$ Find the location of second piece at $t=2 \mathrm{s}.$

Three man $A,B$ and $C$ of mass $40 \mathrm{kg}, 50 \mathrm{kg}$ and $60 \mathrm{kg}$ are standing on a plank of mass $90 \mathrm{kg}$, which is kept on a smooth horizontal plane. If $A$ and $C$ exchange their positions then mass $B$ will shift:-

Two masses $m_1=4\mathrm{kg}$ and $m_2=2\mathrm{kg}$ are released from rest as shown in figure. 

Find the distance $\left(\mathrm{in} \mathrm{m}\right)$  travelled by centre of mass in $9 \mathrm{s}$.