Motion In A Circle (Copy)

A2 Level Physics – Section 12: Circular Motion (Detailed Notes)

12.1 Kinematics of Uniform Circular Motion

1. Angular Displacement and the Radian

• Angular displacement (θ): the angle through which an object moves on a circular path.
• Radian is the SI unit of angular displacement.
• Definition: 1 radian is the angle subtended at the center of a circle by an arc of length equal to the radius.

Conversion:

• 1 revolution = 360° = 2π radians
• θ (in radians) = s / r
  • s = arc length
  • r = radius

2. Angular Speed (ω)

• Angular speed (ω): rate of change of angular displacement.
  ω = θ / t
  • Unit: rad/s
• For one complete revolution:
  θ = 2π radians
  So, ω = 2π / T
  • T = time period (s)

3. Linear Speed in Circular Motion

• Linear speed (v) is the speed along the circular path.
• Related to angular speed:
  v = r·ω
  • r = radius of the circle

12.2 Centripetal Acceleration

1. Nature of Centripetal Force

• In circular motion, velocity changes direction → object is accelerating.
• This acceleration is centripetal (toward the center of the circle).
• Requires a force that is:
  • Constant in magnitude
  • Always perpendicular to velocity
  • Always directed toward the center

2. Centripetal Acceleration Causes Circular Motion

• Even if angular speed is constant, direction of motion continuously changes.
• This requires centripetal acceleration (a) directed toward the center.
• No tangential acceleration in uniform circular motion.

3. Centripetal Acceleration Equations

• Two equivalent forms:

1. a = v² / r
   • v = linear speed
   • r = radius
2. a = r·ω²
   • ω = angular speed
   • r = radius

• Both describe radial (inward) acceleration required to keep the object in circular path.

4. Centripetal Force Equations

• Newton’s second law: F = m·a
• Two equivalent forms:

1. F = mv² / r
2. F = mr·ω²

• F = centripetal force (N)
• m = mass (kg)
• v = linear speed (m/s)
• ω = angular speed (rad/s)
• r = radius (m)

Centripetal force is not a new force—it is the name given to whatever resultant force is causing circular motion (e.g., tension, gravity, friction, etc.)