Lecture on Rotational Motion

Introduction to Rotational Motion

• Definition: Rotational motion refers to any object that can rotate or spin, differing from linear motion where an object moves forward.
• Terminology:
  • Linear motion involves terms like position and displacement, similar to rotational motion which includes angular position and angular displacement.

Angular Position and Displacement

• Angular Position: A point on a circle.
• Angular Displacement: Difference between final and initial angular positions, represented as Δθ (Delta Theta).
• Standard Unit: Radians (though degrees can also be used).

Angular Velocity

• Definition: Measures how fast an object is spinning on a circle.
• Comparison to Linear Velocity: Linear velocity measures how fast an object moves forward.
• Units:
  • Linear velocity: meters per second (m/s).
  • Angular velocity: radians per second (rad/s).
• Formula:
  • Linear velocity (v) = Angular velocity (ω) × Radius (r).

Example: Circle Spin

• Concept: All points on a circle spin at the same angular velocity, but linear velocities vary based on distance from the center.
• Illustration: Point A and Point B on a circle have the same angular velocity but different linear velocities due to differing radii.

Period and Frequency

• Period (T): Time for one complete cycle (seconds per cycle).
• Frequency (f): Number of cycles per second, measured in Hertz (Hz).
• Equations:
  • Angular velocity (ω) = 2πf.
  • ω = 2π/T.
  • Period (T) = 1/f.

Angular and Linear Acceleration

• Linear Acceleration: Change in velocity over time.
• Angular Acceleration: Change in angular velocity over time, measured in radians per second squared (rad/s²).
• Types of Accelerations in Circular Motion:
  • Centripetal Acceleration: Points towards the center of the circle.
    • Formula: a_c = v²/r = ω²r.
  • Tangential Acceleration: Caused by angular acceleration, perpendicular to centripetal acceleration.

Net Acceleration

• Constant Speed: Only centripetal acceleration exists.
• Variable Speed: Both centripetal and tangential accelerations exist.
• Net Acceleration: Vector sum of centripetal and tangential accelerations, forming the hypotenuse of a right triangle.