Angular Variables in Circular Motion

What is the linear velocity if the angular velocity vector is $\overrightarrow{\omega }=3\hat{\mathrm{i}}-4\hat{\mathrm{j}}+\hat{\mathrm{k}}$  and position vector is $\overrightarrow{r}=5\hat{\mathrm{i}}-6\hat{\mathrm{j}}+6\hat{\mathrm{k}}$ ?

Spotlight $\mathrm{S}$ rotates in a horizontal plane with a constant angular velocity of $0.1 \mathrm{rad} {\mathrm{s}}^{-1}$. The spot of light $\mathrm{P}$ moves along the wall at a distance of $3 \mathrm{m}$. The velocity of the spot $\mathrm{P}$ when $\mathrm{\theta }=45°$ (see. fig.) is 

A solid body rotates an angle $\theta$ about a stationary axis according to the law $\theta =6t-2t^3$. What is the mean value of angular velocity (in $\mathrm{rad} {\mathrm{s}}^{-1}$)over the time interval between $t=0$ and the time when the body comes to rest?

The angular displacement of a particle, rotating with constant angular acceleration is $\theta =14 t+4 t^2; \theta$ is measured in radian and $t$ in second. The angular acceleration of the particle is,

How much distance is covered by the tip of a minute hand of length 5 cm when it moves from 3 to 6

Determine the angular speed of a $40 \mathrm{rpm}$ turntable in $\mathrm{rad}/\mathrm{s}.$

A rod of length $L$ is pivoted at one end and is rotated with a uniform angular velocity, in a horizontal plane. Let $T_1$ and $T_2$ be the tensions at points $\frac{L}{4}$ and $\frac{3L}{4}$, away from the pivoted ends.

A particle when moves in a circle with a uniform speed:

A ring of radius $\mathrm{R}$ rolls without slipping on a rough horizontal surface with a constant velocity. The radius of curvature at the highest point of the path followed by any particle on the ring :

A particle $P$ is moving in a circle of radius $\text{'}r\text{'}$ with a uniform speed $v$. $C$ is the centre of the circle and $\mathit{\text{AB}}$ is a diameter. When passing through $B$ the angular velocity of $P$ about $A$ and $C$ are in the ratio

A particle is moving in a circle with constant tangential acceleration. Which one of the following is constant?

A point moves along a circle such that distance travelled $S=a{t}^2$ [where $\mathrm{a}=\frac{1}{2} \mathrm{m}/{\mathrm{s}}^2$] and $t$ is time in second. Find the total acceleration of the point at the moment when it covered the ${\mathrm{n}}^{\mathrm{th}}(\mathrm{n}=0.10)$ fraction of the circle after the beginning of the motion to the closest integer. [Take ${\pi }^2=10$]

One end of a string having length $\left(L\right) 0.5 \mathrm{m}$ attached to the a ball of mass $\left(m\right) 0.5 \mathrm{kg}$ . The ball is rotated on a horizontal circular path about vertical axis. The maximum tension that the string can bear is $324 \mathrm{N}$. If maximum value of angular velocity of ball $\left(\mathrm{i}\mathrm{n} rad{s}^{-1}\right)$ is N then what is the value of $\frac{N}{6}$?

A body of mass $\mathrm{m}=10 \mathrm{kg}$ is attached to one end of a wire of length $0.3 \mathrm{m}.$ The maximum angular speed ( in rad $s^{-1}$ ) with which it can be rotated about its other end in space station is (Breaking stress of wire $=4.8\times {10}^7{\mathrm{Nm}}^{-2}$ and area of cross section of the wire $\left.={10}^{-2} {\mathrm{cm}}^2\right)$ is :

The angle turned by a body undergoing circular motion depends on time as $\theta ={\theta }_0+{\theta }_{1}t+{\theta }_2{t}^2$, Then the angular acceleration of the body is :

The angular speed of a car increases from $600 \mathrm{rpm}$ to $1200 \mathrm{rpm}$ in $10 \mathrm{s}$. What is the angular acceleration of the car?

An object is moving around a circular path of radius $R$ with speed of $v \mathrm{m}/\mathrm{s}$. The value of tangential acceleration in Uniform circular motion is