A cord is wrapped around the rim of a solid cylinder of radius $0.25\mathrm{m}$ and a constant force of $40 \mathrm{N}$ is exerted on the cord. The cylinder is mounted on frictionless bearings and its moment of inertia is $4 \mathrm{kg}-{\mathrm{m}}^2$ The angular velocity of the cylinder after $5\mathrm{m}$ of cord has been unwound, is ${\omega }_1$. If the $40\mathrm{N}$ force is replaced by a $40\mathrm{N}$ weight and released from rest at the top, the angular velocity of the cylinder after $5\mathrm{m}$ of cord has been unwounded is ${\omega }_2$ (Given, $\mathrm{g}=10{ \mathrm{ms}}^{-2}$ ). Then

 

A uniform rod of length $l$ and mass $m$ is free to rotate in a vertical plane about $A$. The rod initially in horizontal position is released. The initial angular acceleration of the rod is (Moment of inertia of rod about $A$ is $\frac{m{l}^2}{3}$)

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 rod of length $50 cm$ is pivoted at one end. It is raised such that it makes an angle of ${30}^o$ from the horizontal as shown and released from rest. Its angular speed when it passes through the horizontal (in $rad s^{-1}$ ) will be $\left(g=10 m{s}^{-2}\right)$

Point masses ${\mathrm{m}}_1$ and ${\mathrm{m}}_2$ are placed at the opposite ends of rigid rod of length $\text{L}$, and negligible mass. The rod is to be set rotating about an axis perpendicular to it. The position of point $\text{L}$ on this rod through which the axis should pass so that the work required to set the rod rotating with angular velocity ${\mathrm{\omega }}_0$ is minimum, is given by: