A uniform square plate of mass $m$ and edge $a$ initially at rest starts rotating about one of the edge under the action of a constant torque $\tau$. Then at the end of the $5^{\text{th }}\sec$ after start

A thin circular coin of mass $5 \mathrm{gm}$ and radius $\frac{4}{3} \mathrm{cm}$ is initially in a horizontal $xy$-plane. The coin is tossed vertically up ($+z$ direction) by applying an impulse of $\sqrt{\frac{\mathrm{\pi }}{2}}\times {10}^{-2} \mathrm{N}-\mathrm{s}$ at a distance $\frac{2}{3} \mathrm{cm}$ from its center. The coin spins about its diameter and moves along the $+z$ direction. By the time the coin reaches back to its initial position, it completes $n$ rotations. The value of $n$ is _____.

[Given: The acceleration due to gravity $g=10 \mathrm{m} {\mathrm{s}}^{-2}$]

A particle of mass $100 \mathrm{g}$ is projected at time $t=0$ with a speed $20 \mathrm{m} {\mathrm{s}}^{–1}$ at an angle $45°$ to the horizontal as given in the figure. The magnitude of the angular momentum of the particle about the starting point at time $t=2 \mathrm{s}$ is found to be $\sqrt{K} \mathrm{kg} {\mathrm{m}}^2 {\mathrm{s}}^{-1}$. The value of $K$ is ______.

(Take $g=10 \mathrm{m} {\mathrm{s}}^{-2}$)

An annular disk of mass $M$, inner radius $a$ and outer radius $b$ is placed on a horizontal surface with coefficient of friction $\mu$, as shown in the figure. At some time, an impulse $J_0\stackrel{\wedge }{x}$ is applied at a height $h$ above the center of the disk. If $h=h_m$ then the disk rolls without slipping along the $x$-axis. Which of the following statement(s) is(are) correct?

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, with what acceleration will the mass fall on release?

A uniform rod of mass $m$, length $\mathcal{l}$ rests on a smooth horizontal surface. Rod is given a sharp horizontal impulse $p$ perpendicular to the rod at a distance $\frac{\mathcal{l}}{4}$ from the centre. The angular velocity of the rod will be

A particle of mass $1 \mathrm{kg}$ is attached to a string of length $5 \mathrm{m}.$ The string is attached to a fixed point $O.$ It is released from the position as shown in figure. Calculate

(a) the impulse developed in the string when it becomes taut;
(b) the velocity of the particle just after the string becomes taut;
(c) the impulse developed in this string $PQ$ at this instant.