For a rigid circular body rotating about its axis with a constant angular speed, the magnitude of the velocity of particles situated at one of its radii:

Figure below shows a shampoo bottle in a perfect cylindrical shape

In a simple experiment, the stability of the bottle filled with different amount of shampoo volume is observed. The bottle is tilted from one side and then released. Let the angle $\theta$ depicts the critical angular displacement resulting in the bottle losing its stability and tipping over. Choose the graph correctly depicting the fraction $f$ of shampoo filled ($f=l$ corresponds to completely filled) vs the tipping angle $\theta$.

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

A wide bottom cylindrical massless plastic container of height $9 \mathrm{cm}$ has $40$ identical coins inside it and is floating on water with $3 \mathrm{cm}$ inside the water. If we start putting more of such coins on its lid, it is observed that after $\mathrm{N}$ coins are put, its equilibrium changes from stable to unstable. Equilibrium in floating is stable if the geometric center of the submerged portion is above the center of mass of the object). The value of $\mathrm{N}$ is closest to

As shown in figure, when a spherical cavity (centred at $O$ ) of radius $1$ is cut out of a uniform sphere of radius $R$ (centred at $C$ ), the centre of mass of remaining (shaded part of sphere is at $G,$ i.e., on the surface of the cavity. $R$ can be determined by the equation:

Three point particles of masses $1.0 \mathrm{k}\mathrm{g}, 1.5 \mathrm{kg}$ and $2.5 \mathrm{k}\mathrm{g}$ are placed at three corners of a right angle triangle of sides $4.0 \mathrm{c}\mathrm{m}, 3.0 \mathrm{c}\mathrm{m}$ and $5.0 \mathrm{c}\mathrm{m}$ as shown in the figure. The centre of mass of the system is at a point:

A wheel of radius "rolls without slipping with a speed $v$ on a horizontal road. When it is at a point $A$ on the road a small bob of the mud separates from wheel at the highest point and touches the point $B$ on the road as shown in the figure. Then $AB$ is (g-acceleration due to gravity)

A rigid massless rod of length $3l$ has two masses attached at each end as shown in the figure. The rod is pivoted at point $P$ on the horizontal axis. When released from the initial horizontal position, its instantaneous angular acceleration will be

In the figure $A$ and $B$ are identical blocks, each of mass $m$ and $C$ is a circular disc of equal mass $m$. Pulley is massless and ffictionless and thread is inextensible. Neglecting friction, find the acceleration of block $A$ and $B$.

A light thread has been tightly wrapped around a disc of mass $M$ and radius $R$. The disc has been placed on a smooth table, lying flat as shown. The other end of the string has been attached to a mass $m$ as shown. The system is released from rest. Find the acceleration of the disc and block.

A uniform rod rotates in a vertical plane about the shown axis. Angular velocity at the moment is $\omega$ . Then, force on the rod by the axis at the moment