Kinematics of Rotation

Angular position of a line of a disc of radius$r=6 \mathrm{cm}$ is given by $\theta =10-5t+4t^2$. The average angular speed (in $\mathrm{rad} {\mathrm{s}}^{-1}$) between $1 \mathrm{s} \mathrm{and} 3 \mathrm{s}$ is:

$\theta =2t^2+3t+1$

Find ${\omega }_{av}$ in $2$seconds

Find the maximum height upto which disc can reach on the incline plane when given velocity $v$ horizontally and disc is performing pure rolling motion.

. A coin placed on a revolving disc, with its centre at a distance of$6 \mathrm{cm}$from the axis of rotation just slips off when the speed of the revolving disc exceeds$45 \mathrm{r}.\mathrm{p}.\mathrm{m}$. What should be the maximum angular speed of the disc, so that when the coin is at a distance of $12 \mathrm{cm}$ from the axis of rotation, it does not slip?

A rod of length $1 \mathrm{m}$ is sliding in a corner as shown. At an instant when the rod makes an angle of $60°$ with the horizontal plane, the velocity of point $A$ on the rod is $1 \mathrm{m} {\mathrm{s}}^{-1}$. The angular velocity of the rod at this instant is:

the angular speed of earth around the Sun is 

The velocity of the centre of mass of a solid sphere of radius $R$rotating with angular velocity about an axis passing through its centre of mass is

A solid sphere is rolling down an inclined plane without slipping. If the inclined plane has inclination $\theta$ with the horizontal, then the coefficient of friction between the sphere and the inclined plane should be

Relation between angular acceleration and linear acceleration.

A block of mass $M$ is released from rest on top of a smooth inclined plane of inclination$\theta$. The block is connected to a light string which is wound over a disc shaped pulley of mass $M$ and radius$R$ as shown. If length of inclined surface is $I$, then time (in second) taken by the block to reach the bottom is-------------------(given:$\frac{l}{g\sin \theta }=3, in SI units$)

When a ceiling fan is switched off, its angular velocity reduces to $50\%$ of its initial value while it makes $36$ rotations. If angular retardation of the
fan is uniform, then how many more rotations will it make before coming to rest?
 

A particle of mass$m$ is whirled in horizontal circle with the help of two threads of length$l$ each as shown in figure. Angular velocity equals $\omega$, then

The equation for angular displacement of a particle, moving in a circular path (radius$200 \mathrm{m}$) is given by:$\theta =18t+3t^2-2t^3$ where $\theta$ is the angular displacement at the end of$t \sec$. Find (i) angular velocity and acceleration at start, (ii) time when the particle reaches its maximum angular velocity, and(iii) maximum angular velocity of the particle.

A disc of radius $R$ is rolling without slipping with velocity $v_0$ on a stationary surface. The radius of curvature of path of particle B at the position shown is B

The speed of a particle in a circle is $2 \mathrm{m} {\mathrm{s}}^{-1}$ along the circumference of a circle of radius $0.5 \mathrm{m}$ If the speed increases at the rate of $6 \mathrm{m} {\mathrm{s}}^{-2}$. The acceleration of the particle at a given instant  is 

Find the height at which the string becomes slack

A wheel of radius $1 \mathrm{m}$ is spinning with angular velocity which varies with time as $5 t^2 \mathrm{rad}/\mathrm{s}$. The acceleration of the point on the wheel's rim at $t=1 \mathrm{s}$, has the magnitude equal to _____.

In $7\mathrm{s}$ a car accelerates uniformly form rest to such a speed that its wheels are turning at a rate of $6 r\omega /s$. How many revolutions does the wheel turn?

A particle is moving with constant speed $v$ along $x$-axis in positive direction. Find the angular velocity of the particle about the point $\left(0,b\right)$, when position of the particle is $\left(a,0\right)$

A particle is revolving about an axis with angular speed of $5\pi$ $\mathrm{rad} {\mathrm{s}}^{-1}$. It comes to rest after $2.5$ seconds. Angular speed of the particle after $2$ seconds from the start is?