Kinematics Of Uniform Circular Motion (Copy)

Radian and Angular Displacement

• Definition of Radian:
  • A radian is the SI unit for measuring angles.
  • 1 radian is defined as the angle subtended at the centre of a circle by an arc length equal to the radius of the circle.
  • Mathematically:
    • If arc length = s and radius = r, then angle in radians θ = s / r
• Relationship Between Radians and Degrees:
  • One complete revolution = 360°
  • One complete revolution also = 2π radians
  • Therefore:
    • 180° = π radians
    • 1 rad = 180° / π ≈ 57.3°
    • 1° = π / 180 radians ≈ 0.01745 radians
• Conversion Examples:
  • Convert 90° to radians:
    • 90° × (π / 180) = π/2 radians
  • Convert π radians to degrees:
    • π × (180 / π) = 180°
• Angular Displacement (θ):
  • Angular displacement is the angle (in radians) that a rotating body moves through.
  • It is a vector quantity, with direction (e.g., clockwise or anticlockwise).
  • Measured in radians (rad).
  • For an object moving along a circular path of radius r:
    • θ = s / r
    • where s = arc length (linear displacement along the circle)

Concept of Angular Speed

• Angular Speed (ω):
  • Defined as the rate of change of angular displacement.
  • Symbol: ω (omega)
  • Unit: radians per second (rad/s)
  • Formula:
    • ω = θ / t
    • where θ = angular displacement, t = time
• Uniform Circular Motion:
  • When a particle moves in a circular path at a constant speed, it still accelerates due to continuous change in direction.
  • Angular speed ω remains constant in uniform circular motion.
• Relation Between Angular Speed and Frequency:
  • One complete revolution = 2π radians
  • If the object completes f revolutions per second, then:
    • ω = 2πf
• Relation Between Angular Speed and Time Period:
  • Time period (T) = time taken for one complete revolution
  • Frequency (f) = 1 / T
  • Hence:
    • ω = 2π / T

Linear Speed and Its Relation with Angular Speed

• Linear Speed (v):
  • This is the speed at which the object moves along the circumference of the circle.
  • Measured in m/s
  • Even though angular speed remains the same for all particles in a rigid rotating object, linear speed varies depending on radius.
• Relation between Linear Speed and Angular Speed:
  • v = rω
    • where:
      • v = linear speed (m/s)
      • r = radius of circular path (m)
      • ω = angular speed (rad/s)
• Example:
  • A point on the edge of a rotating disc of radius 0.3 m has an angular speed of 10 rad/s.
  • Linear speed = rω = 0.3 × 10 = 3 m/s

Key Equations in Circular Motion Kinematics

Important Points and Concepts

• Although the object moves at constant speed in uniform circular motion, it has a centripetal acceleration because the direction of velocity continuously changes.
• Linear speed is tangential to the circular path at every point.
• Angular speed is the same for all points on a rigid rotating object, but linear speed increases with distance from the centre.

Applications in Real World

• Car turning on a curved track:
  • Tyres closer to the centre of the turn have a smaller radius and hence lower linear speed but same angular speed.
• CD/DVD spinning:
  • The centre moves slower in linear terms than the outer edge, but angular speed is constant.
• Astronomy:
  • Planets in circular orbits exhibit uniform circular motion with constant angular velocity (neglecting eccentricity).

Units Recap

• Angular displacement (θ): radians (rad)
• Angular speed (ω): radians per second (rad/s)
• Linear speed (v): metres per second (m/s)
• Time period (T): seconds (s)
• Frequency (f): Hertz (Hz)

Sample Questions and Examples

• Q1: A wheel completes 3 revolutions in 2 seconds. What is the angular speed?
  • One revolution = 2π rad
  • Total angle = 3 × 2π = 6π rad
  • Time = 2 s
  • ω = θ / t = 6π / 2 = 3π rad/s ≈ 9.42 rad/s

• Q2: An object moves in a circle of radius 0.5 m at an angular speed of 4 rad/s. What is its linear speed?
  • v = rω = 0.5 × 4 = 2 m/s

• Q3: The angular speed of a fan is 6.28 rad/s. Find the time period of rotation.
  • ω = 2π / T ⇒ T = 2π / ω = 2π / 6.28 = 1 s

• Q4: A rotating body covers an angular displacement of 5π radians in 10 seconds. What is its angular speed?
  • ω = θ / t = 5π / 10 = 0.5π rad/s ≈ 1.57 rad/s

Conceptual Difference: Angular vs Linear Motion

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