An uniform rod $AB$ having length $L=1.8 \mathrm{m}$ and mass $M$ is pivoted at the center$O$ in such a way that it can rotate freely in the vertical plane. The rod is initially in the horizontal plane as shown in figure. An insect$S$ of the same mass $M$ falls vertically with speed$v$on the point$C$, which lies in middle between the points$O$and$B$. After falling, the insect walks towards the end$B$such that the rod keeps rotating with a constant angular velocity $\omega$.



When the insect reaches the end B, it was seen that the rod has turned through an angle of $\frac{\mathrm{\pi }}{2}$, find the value of $v$ in $\frac{m}{s}$.

Persons $A$ and $B$ are standing on the opposite sides of a $3.5 \mathrm{m}$ wide water stream that they wish to cross. Each one of them has a rigid wooden plank whose mass can be neglected. However, each plank is only slightly longer than, $3 \mathrm{m}.$ So, they decide to arrange them together as shown in the figure schematically. With $B$ (the mass $17 \mathrm{kg}$ ) standing, the maximum mass of $A$, who can walk over the plank is close to,

A uniform bar of mass $M$ is supported by a pivot at its top about which the bar can swing like a pendulum. If a force $F$ is applied perpendicular to the lower end of the bar as shown in figure, what is the value of $F$ in order to hold the bar in equilibrium at an angle $\left(\theta \right)$ from the vertical

A particle of mass $2 \mathrm{kg}$ is on a smooth horizontal table and moves in a circular path of radius $0.6 \mathrm{m}$. The height of the table from the ground is $0.8 \mathrm{m}$. If the angular speed of the particle is $12\mathrm{ }\mathrm{rad}\mathrm{ }{\mathrm{s}}^{-1}$ , the magnitude of its angular momentum about a point on the ground right under the center of the circle is:

A wheel of radius, $R$ with an axle of radius, $\frac{R}{2}$ is as shown in the figure and is free to rotate about a frictionless axis through its center and perpendicular to the page. Three forces $\left(F, F, 2 F\right)$, are exerted tangentially to the respective rims as shown in the figure.

The magnitude of the net torque acting on the system is nearly,