Circular Motion – Complete Notes
Introduction to Circular Motion
When a body moves along a circular path with uniform speed, its motion is called uniform circular motion. Although the speed is constant, the velocity changes continuously because the direction of motion changes at every point on the circular path.
📌 Key Point: In uniform circular motion, speed is constant but velocity is not — because direction changes continuously.
Angular Quantities
Angular Displacement (θ)
Angular displacement is the angle subtended at the center of the circular path by the arc traced by the moving body. It is measured in radians (rad).
θ = Arc length / Radius = s / r
Angular Velocity (ω)
Angular velocity is the rate of change of angular displacement. It is the angle swept per unit time and is measured in radians per second (rad/s).
ω = θ / t = 2π / T = 2πf
Definition: Time Period (T): The time taken to complete one full revolution is called the time period.
Centripetal Force
The force directed towards the center of the circular path that keeps the body moving in a circle is called centripetal force. Without this force, the body would move in a straight line (inertia).
F_c = mv² / r = mω²r = m(4π²r / T²)
Examples of centripetal force:
- ✓Gravitational force for satellite orbiting Earth
- ✓Tension in string for a ball whirled in a circle
- ✓Normal force + friction for a car on a curved road
- ✓Electrostatic force for electron orbiting nucleus
Banking of Roads
When a road is tilted at an angle θ inward, it is called a banked road. Banking reduces the wear on tires and allows vehicles to take turns at higher speeds without relying solely on friction.
tan θ = v² / rg
📝 Solved Example
Example: A car of mass 1000 kg moves with a speed of 20 m/s on a circular road of radius 100 m. Find the centripetal force acting on the car. Solution: F_c = mv²/r = (1000 × 20²) / 100 = 400,000 / 100 = 4000 N
Solved Numericals
A body of mass 2 kg moves in a circle of radius 0.5 m with angular velocity of 4 rad/s. Find: (a) linear velocity (b) centripetal acceleration (c) centripetal force.
📝 Solved Example
Given: m = 2 kg, r = 0.5 m, ω = 4 rad/s (a) v = ωr = 4 × 0.5 = 2 m/s (b) a_c = v²/r = (2)²/0.5 = 4/0.5 = 8 m/s² OR: a_c = ω²r = 16 × 0.5 = 8 m/s² (c) F_c = ma_c = 2 × 8 = 16 N