Topic 31 (1)

Topic 3-1: Circular Motion

Motion in two dimensions can be understood through circular motion, using polar coordinates where the radius (R) is constant while only the angle (θ) varies over time.

Angular Kinematics:

• Angular position: θ(t) (in radians/degrees)

• Angular velocity: ω(t) = dθ(t)/dt

• Angular acceleration: α(t) = dω(t)/dt = d²θ(t)/dt²

• Units: ω (rad/s, deg/s), α (rad/s², deg/s²)

Newton's Laws for circular motion:

• Translation: Movement of center of mass along a circle.

• Rotation: Object spins about its center.

• 2nd Law: ΣF = ma (linear acceleration). In circular motion, this includes centripetal (toward center: a_c = v²/R) and tangential (along motion: a_t = dv/dt) accelerations.

Relationship between Linear and Angular Kinematics: Distance traveled (arc length) is s = Rθ. Tangential velocity: v_t(t) = d(s)/dt. Acceleration: a_c(t) = Rω², a_t(t) = Rα.

Uniform Circular Motion (UCM): Constant speed with only centripetal acceleration. Period (T) and frequency (f) link linear and angular speed: T = 2πR/v = 2π/ω, ω = 2πf.

Examples:

1. Total linear acceleration at t=10s given specific ω and α.

2. Speed ratio of satellites based on their orbital radii (v1/v2 = √(R2/R1)).