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 smooth semicircular wire track of radius $R$ is fixed in a vertical plane. One end of a massless spring of natural length  $\frac{3\mathrm{R}}{4}$ is attached to the lowest point $O$ of the wire track. A small ring of mass $m$ which can slide on the track is attached to the other end of the spring. The ring is held stationary at point $P$ such that the spring makes an angle of $60°$ with the vertical. The spring constant $K=\frac{mg}{R}$. Consider the instant when the ring is released. The normal reaction on the ring by the track is

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.

In the figure shown a smooth ring is connected to rod $AB$, while rod $CD$ passes through ring. At the given instant angular velocity of rod $AB$ about hinge $A$ is $1\mathrm{rad} {\mathrm{s}}^{-1}$ and $AC=CB$ instantaneous angular velocity of rod $CD$ about hinge $C$ is( in $\mathrm{rad} {\mathrm{s}}^{-1}$)

A circular cylinder of radius $R$ and height $H$ is filled with water to a height $\frac{2}{3}H.$ It starts rotating about its axis with angular speed $\omega ,$ which is increased very slowly. Choose the correct statement(s). (Assume liquid is rotating along with the vessel)

A particle describes a horizontal circle in a conical funnel whose inner surface is smooth with the speed of $1 \mathrm{m} {\mathrm{s}}^{-1}$ . What is the height (in $\mathrm{cm}$) of the plane of the circle from the vertex of the funnel? $\left(g=10 \mathrm{m} {\mathrm{s}}^{-2}\right)$

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?

A small object is placed on a smooth table at a distance $r_0$ from a hole in the centre of table. An ideal inextensible chord is attached to the object through this hole. The object is set into motion with initial velocity of magnitude $v_0$ at right angle to the chord and at the same time, the chord is pulled through the hole at uniform speed ${\mathrm{cms}}^{-1}$. Initially object is at point $A\left(0,r_0\right)$ and at any time $t$, it is at point $P\left(r\cos \theta ,r\sin \theta \right)$ Neglect the dimension of object. Choose CORRECT option(s)

A point moves with deceleration along the circle of radius $R$ so that at any time its tangential and normal acceleration are equal in magnitude. At the initial moment $t=0$, the velocity of the point is $v_0$. The velocity of the point during motion will be

A solid body rotates about a stationary axis so that its angular velocity depends upon rotation angle $\theta$ as $\omega ={\omega }_0-K\theta$ where ${\omega }_0$ and $\theta$ are positive constant. Which of the following variation are correct?

A uniform sphere of radius$R$  is placed on a rough horizontal surface and given a linear velocity $v$ and angular velocity $\omega$ as shown. The sphere comes to rest after moving some distance to the right. It follows that $v=x\omega R$ where $x$ is

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 $\mathrm{rad}/\mathrm{s}.$