Consider three systems, each with three masses, ${\mathrm{m}}_1, {\mathrm{m}}_2, {\mathrm{m}}_{\mathit{3}}\left({\mathrm{m}}_{\mathit{1}}>{\mathrm{m}}_2>{\mathrm{m}}_3\right)$ with centre of mass at point $A\mathit{,}$ as shown in the figure - $1,$ figure - $2$ and figure -$3$ respectively. The same force $\overrightarrow{\mathit{F}}$ is applied on the each system separately to just one of the three masses as shown. Which of the following statements is true. 

  

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

Two particles of masses ${\mathrm{m}}_1,\mathrm{ }{\mathrm{m}}_2$ move with initial velocities ${\mathrm{u}}_1$ and ${\mathrm{u}}_2$. On collision, one of the particles get excited to higher level, after absorbing energy $\mathrm{\epsilon .}$ If final velocities of particles be ${\mathrm{v}}_1$ and ${\mathrm{v}}_2$ then we must have:

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.

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

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 :

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: