A circular hole of radius $\left(\frac{a}{2}\right)$ is cut out of a circular disc of radius $a$ as shown in figure. The centroid of the remaining circular portion with respect to point $O$ will be:

A sphere of radius $a$ and mass $m$ rolls along a horizontal plane with constant speed $v_0$. It encounters an inclined plane at angle $\theta$ and climbs upward. Assuming that it rolls without slipping, how far up the sphere will travel?

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 :

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)

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:

A uniform cylinder of mass $M$ and radius $R$ is to be pulled over a step of height $a\left(a<R\right)$ by applying a force $F$ at its centre $\text{'}\mathrm{O}\text{'}$ perpendicular to the plane through the axes of the cylinder on the edge of the step (see figure). The minimum value of $F$ required is: