Pure Rotational Motion

A disk of mass $2 \mathrm{kg}$ and diameter $20 \mathrm{cm}$ is rotating about an axis passing through its center perpendicular to its plans with an angular speed that varies with time as $2t^2+3t+2$, find the torque experienced by the disk at the instant $t=2 \mathrm{s}$.

A light thread is wound on a disk of mass $m$ and other end of the thread is connected to a block of mass $m$, which is placed on a rough ground as shown in the diagram. Find the minimum value of the coefficient of friction for which the block remains at rest:

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A uniform thin circular ring of mass $m\left(m=0.4 \mathrm{kg}\right)$ and radius $R$ has a small particle of the same mass $m$ fixed on it as shown in the figure. The line joining the particle to centre is initially horizontal. The ground is frictionless. Find the contact force (magnitude) exerted by the ground on the ring, when the system is released from rest.

Two blocks of masses $2 m$ and $3 m$ are connected through a string which passes over a pulley (disc) of mass $m$ and radius $R$ as shown in figure. If system is released from rest and string does not slip on the pulley, then angular acceleration of the pulley is

A uniform rod of length $2l$ and mass $4\mathrm{m}$ is released to rotate about hinged end $A$ in vertical plane as shown in figure. The magnitude of initial angular acceleration of rod is

If a constant torque of $500 \mathrm{N} \mathrm{m}$ turns a wheel of moment of inertia  $100 \mathrm{kg} {\mathrm{m}}^2$  about an axis through its centre, find the angular velocity and the kinetic energy gained in $2$ seconds.

The flywheel of an engine when acted upon by a constant torque rotates at the rate of $3000 \mathrm{rev} {\min }^{-1}$ and develops a power of $10 \mathrm{kW}$. Find the torque acting on the wheel?

Show that the torque acting on a body is equal to the product of moment of inertia and the angular acceleration of the body.

Using the S.I. units of torque and angular acceleration show that the S.I. unit of moment of inertia is $kg m^2$.

A rigid body is rotating. Is it necessary that a net torque acts on the body ?

Show that the work done to rotate a body is equal to the change in rotational kinetic energy.

A couple of $10 \mathrm{N} \mathrm{m}$ is applied to a fly wheel of mass $10 \mathrm{kg}$ and radius of gyration $50 \mathrm{cm}$. What is the resultant angular acceleration?  

A cylinder of mass $M$ is placed on top of a block of mass $2M$. Neglecting the friction between the block and the horizontal plane. But friction coefficient between the cylinder and the top surface of block is $\mu$. A small body of mass $M$ travelling with a velocity $v_0$ hits the block horizontally. Collision is completely elastic. If the maximum value of $v_{\mathit{0}}$ for which the cylinder starts pure rolling before it topples from the top of the block is ${\left(v_0\right)}_{\max }=n\sqrt{m\mu gd}$. The value of $m\times n$ is ____

The $L$ shaped uniform rod having identical limbs each of mass $M$ and length $L$ is pivoted smoothly at point $A$ as shown in figure. When the rod is released with rest from the given position, it swings in a vertical plane. The initial angular acceleration of rod is

A solid sphere rolls on a rough horizontal surface with a linear speed $7 \mathrm{m} {\mathrm{s}}^{-1}$ collides elastically with a fixed smooth vertical wall. Then the speed of the sphere when it has started pure rolling in the backward direction in $\mathrm{m} {\mathrm{s}}^{-1}$ is.

Write a short note on relation between power and torque in totational motion.

A turbine fan in a jet engine has moment of inertia $2.5 \mathrm{kg} {\mathrm{m}}^2$ about its axis of rotation. Its angular velocity is $40t^2$. The net torque is

If $I=50 \mathrm{kg}-{\mathrm{m}}^2$, then the magnitude of torque to be applied to stop it in $10 \mathrm{s}$ is.... . Given, its initial angular speed is $20 \mathrm{rad} {\sec }^{-1}$.

The angular velocity of a body is $\overrightarrow{\omega }=2\hat{\mathrm{i}}+3\hat{\mathrm{j}}+4\hat{\mathrm{k}}$ and a torque $\overrightarrow{\tau }=\hat{\mathrm{i}}+2\hat{\mathrm{j}}+3\hat{\mathrm{k}}$ acts on it. The rotational power (in watt) will be :-

A uniform rod $AB$ of length $l$ and mass $m$ is free to rotate about point $A$. The rod is released from rest in the horizontal position. Given that the moment of inertia of the rod about $A$ is $\frac{m{l}^2}{3}$ the initial angular acceleration of the rod will be :-