Differentiate between centre of mass and centre of gravity.

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 bead of mass $\mathrm{m}$ stays at point $\mathrm{P}\left(\mathrm{a}, \mathrm{b}\right)$ on a wire bent in the shape of a parabola $y=C{x}^{\mathit{2}}$ and rotating with angular speed $\mathrm{\omega }$ (see figure). The value of $\omega$ is (neglect friction)

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

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

A uniform rod of length $200 \mathrm{cm}$ and mass $500 \mathrm{g}$ is balanced on a wedge placed at $40 \mathrm{cm}$ mark. A mass of $2 \mathrm{kg}$ is suspended from the rod at $20 \mathrm{cm}$ and another unknown mass $m$ is suspended from the rod at $160 \mathrm{cm}$ mark as shown in the figure. Find the value of $m$ such that the rod is in equilibrium. $\left.(g=10 \mathrm{m} {\mathrm{s}}^{-2}\right)$

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 :

Shown in the figure is rigid and uniform one meter long rod $\mathrm{AB}$ held in horizontal position by two strings tied to its ends and attached to the ceiling. The rod is off mass $\text{'}\mathrm{m}\text{'}$ and has another weight of mass $2 \mathrm{m}$ hung at a distance of $75 \mathrm{cm}$ from $\mathrm{A}$. The tension in the string at $\mathrm{A}$ is:

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:

A mass $m$ is supported by a massless string wound around a uniform hollow cylinder of mass $m$ and radius $R$. If the string does not slip on the cylinder, then with what acceleration will the mass release? (Assume $g=$ acceleration due to gravity)

A rod of length $L$ can rotate about end $0 .$ What is the work done if the rod is rotated by $180°$