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

Two blocks of mass $m_1=10 \mathrm{kg}$ and $m_2=5 \mathrm{kg},$ connected to each other by a massless inextensible string of length $0.3 \mathrm{m}$ are placed along a diameter of a turn table. The coefficient of friction between the table and $m_1$ is $0.5$ while there is no friction between $m_2$ and the table. The table is rotating with an angular velocity of $10$ rad $s^{-1}$ about a vertical axis passing through its centre $O$. The masses are placed along the diameter of the table on either side of the centre $O$ such that the mass $m_1$ is at a distance of $0.124 \mathrm{m}$ from $O$. The masses are observed to be at rest with respect to an observer on the turn table. Calculate the frictional force on $m_1$.

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 $h$ ?

Three particles, each of mass $m,$ are situated at the vertices of an equilateral triangle of side length $a$. The only forces acting on the particles are their mutual gravitational forces. It is desired that each particle moves in a circle while maintaining the original mutual separation $a$. Find the initial velocity that should be given to each particle and also the time period of the circular motion.

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 

The tangential acceleration of the tip of a blade is $47.13 \mathrm{m}/{\mathrm{s}}^2$ and its centripetal acceleration is $473.9 \mathrm{m}/{\mathrm{s}}^2$. Find the value of linear acceleration of the tip of the blade.

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 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 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