Uniform 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 $\overrightarrow{r}=5\hat{\mathrm{i}}-6\hat{\mathrm{j}}+6\hat{\mathrm{k}}$ ?

A hemispherical bowl of radius $R=0.1 \mathrm{m}$ is rotating about its own axis (which is vertical), with an angular velocity $\omega$. A particle of mass ${10}^{-2} \mathrm{kg}$ on the frictionless inner surface of the bowl is also rotating with the same $\omega$. The particle is at a height $h$ from the bottom of the bowl.
Obtain the relation between $h$ and $\omega$. What is the minimum value of $\omega$ needed, in order to have a non-zero value of $h$ ?

An electric line of force in the $x-y$ plane is given by the equation $x^2+y^2=1$. A particle with unit positive charge, initially at rest at the point $x=1,y=0$ in the $x-y$ plane, will move along the circular line of force.

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 

Why is centrifugal force considered as an imaginary force?

A body of mass $m$ is moving with speed $V$ along a circular path of radius $r$. Now, the speed is reduced to$\frac{V}{2}$  and radius is increased to $3r$. For this change, initial centripetal force needs to he

A particle is moving in a circle of radius $\mathrm{R}$ with a constant speed $\mathrm{v}$. Its average acceleration over the time when it moves over half the circle is :

Two identical particles each of mass $m$ go round a circle of radius $a$ under the action of their mutual gravitational attraction. The angular speed of each particle will be : 

A merry-go-round rotates from rest with constant angular acceleration $\alpha$. Ratio of time to rotate first 2 revolutions and next 2 revolutions is

A child of mass $5 \mathrm{kg}$ is going round a merry-go-round that makes $1$ rotation in $3.14 \mathrm{s}$. The radius of the merry-go-round is $2 \mathrm{m}$. The centrifugal force on the child will be

A particle is moving with constant speed in a circular path. When the particle turns by an angle $90°,$ the ratio of instantaneous velocity to its average velocity is $\pi :x\sqrt{2}.$ The value of $x$ will be

A disc of radius $R$rolls on a rough horizontal surface. The distance covered by the point A in one revolution is:

Vikas is driving a car of mass m along a circular path of radius$R.$ The coefficient of static friction between the path and tyres of the car is ${\mu }_s$ .The maximum speed with which he can drive the car without slipping is ($g$ is acceleration due to gravity)

A particle is moving on a spiral path as shown in the figure. The speed of particle remains constant.

In circular motion speed depends on angular position as $v= \left(\theta +5\right) \mathrm{m}/\mathrm{s}$. The value of $\theta$ as a function of time $t$ is

A phonograph turn-table rotating at $78 \mathrm{rev}/\min$ slows down and stops in $30 \sec$ after the motor is turned off. Then the revolutions made by it in this time are