Circular Motion

The study of circular motion is not only important in itself, but also an essential pre-condition for the study of rotational motion later under Mechanics. We start with the definitions of the angular quantities in circular motion.

Suppose a particle is moving in a circle in the counterclockwise direction. The centre of the circle coincides with the origin O of the coordinate system Oxy. The initial position of the particle is P1 at time t1. The final position of the particle is P2 at a later time t2 . The angular displacement of the particle is equal to the angle swept by its position vector as it moves from OP1 to OP2 . Angular displacement is a pure number without dimensions. However, it is customary to express it in an artificial supplementary SI unit called radian (rad).

The angular velocity of the particle is its angular displacement per unit time. In uniform circular motion, the angular velocity has the same value at every instant. In nonuniform circular motion, the angular velocity keeps changing from one instant to another. The dimension of angular velocity is [T-1] ; its unit is radian per second (rad/s) or simply per second ( /s) since radian is a dimensionless unit. The time taken by a particle to complete one revolution is known as its time period. The number of revolutions completed by the particle per second is called its frequency.

In the case of nonuniform circular motion, the angular acceleration of a particle is the rate of change in angular velocity with time.The dimension of angular acceleration is [T-2] ; its unit is radian per second square (rad /s2), or just per second square ( /s2).

It is useful to note the relations between linear and angular quantities. A particle, moving in a circle of radius r, changes its position from P1 at time t1 to P2 at time t2 . The distance travelled by the particle is the length of the arc P1P2 . This distance is equal to the product of its angular displacement and radius of the circle. Similarly, the linear speed of a particle moving in a circle at any instant is the product of its angular speed and circle's radius.

The linear acceleration of a particle in circular motion has two components: tangential and radial. The tangential acceleration at any instant is the product of its angular acceleration and circle's radius. The radial acceleration, or centripetal acceleration as it is more commonly called, is the product of angular speed squared and circle radius.

Circular motion of a particle with constant angular acceleration is analogous to motion in a straight line with constant linear acceleration. Therefore, it is possible to develop a set of kinematic equations of circular motion in the same way as we did it for rectilinear motion. These equations involve quantities such as angular displacement, initial and final angular velocities, and angular acceleration.

When a particle performs uniform circular motion, its angular acceleration is zero, which means its tangential acceleration is also zero. The only acceleration present is the centripetal acceleration directed radially towards the centre of the circle. According to Newton's second law, the particle experiences a net external force – directed radially inwards – of magnitude given by the product of the mass of the particle and its centripetal acceleration. This force is known as centripetal force. An example of centripetal force is the gravitational force exerted by the sun on the earth which makes the latter revolve round the sun. In a rotating non-inertial frame of reference, a particle experiences a pseudo force directed radially outwards. This pseudo force is known as centrifugal force.

When a particle performs nonuniform circular motion, its tangential acceleration and centripetal acceleration are both non-zero quantities. Therefore, by Newton's second law, the net external force acting on the particle is the resultant of a tangential force and a centripetal force. A sphere moving in a vertical circle at the end of a light string constitutes a case of nonuniform circular motion.

Basic level videos

Angular Quantities In Circular Motion 1:11:16 Basic
350 7.5
Derivation Of Centripetal Acceleration 42:19 Basic
250 4
Two Accelerations Of Nonuniform Circular Motion 53:04 Basic
300 7.5
Relations Between Linear And Angular Quantities 38:28 Basic
200 3
Circular Motion With Constant Angular Acceleration 31:51 Basic
200 3
Problems On Circular Motion With Constant Angular Acceleration 32:47 Basic
200 3
Examples Of Uniform Circular Motion 42:49 Basic
250 4
Examples Of Uniform Circular Motion II 47:42 Basic
300 5
Examples Of Nonuniform Circular Motion 1:04:17 Basic
350 7.5
Problems On Circular Motion I 33:45 Basic
200 3
Problems On Circular Motion II 57:52 Basic
300 5
Problems On Circular Motion III 34:51 Basic
200 3

Advanced level Videos Note: (CE) Stands for Problems from Competitive Examination Papers

Advanced-Level Problems On Circular Motion I 1:03:41
350
7.5
Advanced-Level Problems On Circular Motion II 1:11:15
350
7.5
Advanced-Level Problems On Circular Motion III 1:18:10
350
7.5