Uniform Circular Motion

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.

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:

Two spheres of equal masses are attached to a string of length $2 \mathrm{m}$ as shown in figure. The string and the spheres are then whirled in a horizontal circle about $O$ at a constant rate. What is the value of the ratio of the tension in the string between $P$ and $Q$ with the tension in the string between $P$ and $O$?

A stone tied to one end of rope and rotated in a circular motion. If the string suddenly breaks, then the stone travels

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 :

Starting form rest, a particle rotates in a circle of radius $\text{R}=\sqrt{2}$ m with an angular acceleration α = π/4 rad/s². The magnitude of average velocity of the particle over the time it rotates quarter circle is

In a carnival ride the passengers travel in a circle of radius 5.0 m, making one complete circle in 4 s. What is the acceleration?

The angular velocity of the minute hand of a clock is :

Which of the following statements about the uniform circular motion is true?

To enable a particle to describe circular motion the angle between its velocity and acceleration is given by

The speed of revolution of a particle performing circular motion is halved and its angular speed is doubled. The centripetal acceleration becomes:

A wheel is rotated about a horizontal axle at a constant angular speed. Next it is rotated in the opposite direction with the same angular speed. The acceleration at a point on the top of the wheel in the second case as compared to the acceleration in the first case :

Which one is the correct relation between the magnitude of linear acceleration and angular acceleration in circular motion of radius $R$ of circular path.
Assume $a_t$ is linear acceleration and $\alpha$ is angular acceleration.