Dynamics of Rigid Bodies with Fixed Axis of Rotation

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 $g=10 \mathrm{m} {\mathrm{s}}^{-2}$]

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 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

Find the angular velocity of a body rotating with an acceleration of $2 \mathrm{rev} {\mathrm{s}}^{-2}$ as it completes the $5^{\mathrm{th}}$ revolution after the start.

A body rotates about a fixed axis with an angular acceleration of $1 \mathrm{rad} {\mathrm{s}}^{-2}$. Through what angle does it rotate during the time in which its angular velocity increases from $5 \mathrm{rad} {\mathrm{s}}^{-1}$ to $15 \mathrm{rad} {\mathrm{s}}^{-1}$?

A rod of mass $m$ and length $l$ is suspended from a vertical wall and kept horizontal by a massless vertical thread as shown. The tension in the thread is,

A body is in pure rolling motion on a horizontal surface. For which body the rotational kinetic energy is not less than translational kinetic energy

A constant torque acting on a uniform circular wheel changes its angular momentum from $A_0$ to $4 {\mathrm{A}}_0$ in $4 \mathrm{s}$. The magnitude of this torque is

In a mess fighting match between Bheema and Duryodhana, Bheema swings his mess to hit the target with a constant angular velocity ${\omega }_0$ After hitting target mess comes to rest and surprisingly bheema feels no impulsive reaction on his hand. Mess hits the target at point $\text{'}P\text{'}$ as shown in figure. If Bheema holds the mess at end $\mathrm{A},$ then $2\ell$ will be. $($Given: Moment of inertia of mess about $A$ is $25 \mathrm{kg} {\mathrm{m}}^2.$ Total mass is $20 \mathrm{kg}.$$)$

A uniform rod of mass $\text{'}m\text{'}$ and length $\text{'}l\text{'}$ lying on a smooth horizontal table is pivoted about one end. An impulse $\text{'}J\text{'}$ is applied to the other end. Angular speed with which rod starts rotating is

A mass $m=2 \mathrm{kg}$ is suspended through a thread wrapped around a disc-shaped pulley of mass $6 \mathrm{kg}$ as shown in the figure. Friction in the axle of the pulley is absent and there is no slipping of thread. When the mass $m$ falls through a vertical distance 2 metre, the velocity acquired by it, is $\left(g=10 \mathrm{m}/{\mathrm{s}}^2\right)$

A thin uniform rod of length $l$ and mass $m$ suspended vertically is free to rotate about a smooth horizontal axis which passes through its end $A$. $A$ point mass $m$ moving horizontally with velocity $v_0$ hits the lower end $B$ of the rod and sticks to it. After collision the rod just reaches the horizontal position. Which of the following statements is WRONG?

A rod of mass $3 \mathrm{kg}$ and length $1 \mathrm{m}$ is lying along the Y axis such that one of its end is at the origin. An Impulse is given to the rod due to which its end at origin acquire a velocity of $4 i$ and other end acquires a velocity $8 i$. What is magnitude of angular momentum of rod about origin.

A body rolls down an inclined plane then the ratio of translational kinetic energy to the total kinetic energy is :-

A solid cylinder of radius $r$  and mass $m$  is placed with no initial velocity on a moving belt $\text{'}B\text{'}$ . It is placed against a rough wall $\text{'}A\text{'}$ , which restricts the sliding of the cylinder on the belt as shown in the figure. Assuming that the

coefficient of friction at $A$ and $B$ is $\mu (<1)$, the angular acceleration of the cylinder will be

A uniform thin rod of mass $m$  and length $l$  is held horizontally by two vertical strings attached to the two ends. One of the string is cut. Find the angular acceleration soon after it is cut:

A pivoted beam of negligible mass has a mass suspended from one end and an At wood's machine suspended from the other. The frictionless pulley has negligible mass and dimension. Gravity is directed downward and $M_2=3M_3,l_2=3l_1$. Find the ratio $M_1/M_2$ which will ensure that the beam has no tendency to rotate just after the masses are released. 

Figure shows a small wheel fixed coaxially on a bigger one of double the radius. The system rotates about the common axis. The strings supporting $A$ and $B$ do not slip on the wheels. If $x$ and $y$ be the distances travelled by $A$ and $B$ in the same time interval, then

A wheel of moment of inertia $3 \mathrm{kg} {\mathrm{m}}^2$ rotates about an axis passing through its centre and perpendicular to its plane. Its initial speed of rotation is given as $60 \mathrm{rad}/\mathrm{s}$. Due to friction, the wheel comes to rest in $5$ minutes. Find the angular momentum (in $\mathrm{kg} {\mathrm{m}}^2/\mathrm{s}$) of the wheel three minutes before it stops.

A wheel having an initial angular velocity of zero is subjected to uniform angular acceleration. It is observed that in the first $3 \mathrm{s}$, it rotates through an angle ${\theta }_1$ and in the next $3 \mathrm{s}$, it rotates through an additional angle ${\theta }_2$. Hence, compute the ratio ${\theta }_2/{\theta }_1$.