Circular Motion Basics

Think about a stone tied to a string being whirled in a circle, or the motion of the Earth around the Sun, or a motorcycle turning around a curved road in the hills of Nepal. All of these involve circular motion -- one of the most important types of motion in physics.

Uniform Circular Motion

When an object moves along a circular path with constant speed, it is in uniform circular motion. Even though the speed is constant, the velocity changes because the direction keeps changing. This means the object is accelerating.

Key Quantities

Angular displacement (theta): The angle swept by the radius vector, measured in radians.

Angular velocity (omega): The rate of change of angular displacement.

omega = theta / t = 2 pi / T = 2 pi f

where T is the time period and f is the frequency.

Relation between linear and angular velocity:

v = r omega

where v is linear speed and r is the radius of the circular path.

Centripetal Acceleration

The acceleration in uniform circular motion always points toward the centre of the circle. It is called centripetal (centre-seeking) acceleration.

ac = v² / r = r omega²

Centripetal Force

By Newton's second law, a net inward force is required to maintain circular motion:

Fc = mv² / r = m r omega²

This is not a new type of force -- it is provided by tension, gravity, friction, or other existing forces.

Worked Example

A car of mass 1000 kg moves around a circular track of radius 50 m at a speed of 20 m/s. Find the centripetal force acting on the car.

Solution:

• Fc = mv² / r = 1000 x (20)² / 50
• Fc = 1000 x 400 / 50 = 8000 N

This force is provided by the friction between the tyres and the road.

Real-World Examples

• Banking of roads: Roads in hilly areas of Nepal are banked (tilted) so that the normal force provides the centripetal force needed for safe turning
• Washing machine: Uses centripetal force to spin water out of clothes
• Satellites: Gravity provides the centripetal force for orbital motion

Key Takeaways

• Uniform circular motion has constant speed but changing velocity
• Centripetal acceleration is always directed toward the centre
• Centripetal force is provided by real forces like gravity, tension, or friction
• Linear velocity v = r omega