# Circular Motion

• ## Introduction

Circular motion is a common type of motion and is a part of a more general set of curvilinear motions. In such motions, the forces acting on a body have a component directed along the radius of curvature of the curve towards the center of curvature.

When the motion is circular, this force is directed along the center of the circle and is along the radius of the circle.

The specific name given to that component of force is the centripetal force. So, it is a real force.

Uniform circular motion is the motion, where the magnitude of velocity of the rotating object, i.e. $|\vec v|$ is constant. The only component of acceleration is because of the change in direction of the velocity. For example, a stone attached to a string and revolved in a horizontal circle.

The centripetal force in above case is due to the tension in the string. Its magnitude is

${\vec F_{centripetal} = \frac {mv^2}{r} = mr \omega^2}$

This is equal to the tension in the string.

The centripetal acceleration is directed towards the center, and is given by

${\vec a_{centripetal} = \frac {\vec F_{centripetal}}{m} = \frac {v^2}{r} = r \omega^2}$

The weight of stone DOES NOT play any role.

• ## Vertical Circular Motion

In vertical circular motion, the resultant acceleration is a result of 2 accelerations:

1) due to a continuous change in the direction of velocity of the rotating body and

2) due to acceleration due to gravity

At any instant, the gravity vector $\vec g$ is always directed vertically downward. The acceleration due to change in direction is directed along the radius of the circle and is known as the centripetal acceleration.

Thus, the body is under the influence of 2 forces, viz. the tension $T$ and its weight $mg$.

Note the directions of real forces at each of the 4 points $A,B,C$ and $D$.

• ### Analysis from Stone’s Frame of Reference

For 2 cases discussed above (viz. horizontal and vertical circular motions), when you refer to the ground as the frame of reference, the Newton’s 2nd law of motion governs the motion.

However, for the same cases, when observed from the frame of reference of stone (which is stationary from its frame), one has to account for an additional fictitious force, to bring the stone to standstill. This force is known as the centrifugal force.

Thus, we can write,

${\vec T + m \vec g = \vec F_{centrifugal}, \ vertical \ motion}$

and

${\vec T = \vec F_{centrifugal}, \ horizontal \ motion}$

• ## D’Alembert’s Principle and Circular Motion

The principle converts a problem of dynamics into a problem of statics. i.e. now we assume the rotating stone as our frame of reference. By adding the inertial force (which is the centrifugal force), we get,

${\vec T + m \vec g = \frac {m v^2}{r}}$

OR

${\vec T = \frac {mv^2}{r} - m \vec g}$

• ## Tensions at Top and Bottom Positions

At the top,

${T = \frac {mv^2}{r} - mg}$

At the bottom,

${T = \frac {mv^2}{r}+mg}$

Note that the directions of forces have been taken into consideration.

So, the least speed required at the top, s.t. the string does not slack ($T \to 0$) will be $v_{top} = \sqrt {gr}$.

Using the principle of conservation of energy, we can write,

${\frac 1 2 m v^2_{bottom} = \frac 1 2 m v^2_{top} + mg (2r)}$

$r$ is the radius of the circle, as shown.