Learning targets

• I can distinguish translational and rotational motion.
• I can describe linear and angular displacement.
• I can compare linear and angular velocity and acceleration.
• I can relate linear and angular quantities mathematically.
• I can identify combined motion in everyday systems.

1. Translation: movement without turning

In pure translation, every point of a rigid body experiences the same displacement over a given interval. The path may be straight or curved, but the body does not change its orientation.

Linear displacement: Δx = x_f − x_i (m). Average velocity: v = Δx/Δt (m/s). Average acceleration: a = Δv/Δt (m/s²).

2. Rotation: changing orientation

In pure rotation about a fixed axis, every point turns through the same angle, although points at different distances from the axis travel different arc lengths.

Angular displacement: Δθ (radians). Average angular velocity: ω = Δθ/Δt (rad/s). Average angular acceleration: α = Δω/Δt (rad/s²).

One complete revolution equals 2π radians or 360°. Counterclockwise is often defined as positive, but the sign convention must be stated.

3. Connecting linear and angular quantities

At a fixed angular speed, doubling the radius doubles tangential speed. Every point on the same rigid wheel shares ω but not v.

4. Angular acceleration and circular motion

If angular velocity changes, the body has angular acceleration. Even if angular speed is constant, a point moving in a circle has centripetal acceleration because its velocity direction changes.

Tangential acceleration: a_t = rα. Centripetal acceleration: a_c = v²/r = rω², directed toward the axis.

5. Combined motion: rolling

A rolling wheel translates as its center moves forward and rotates about its center. For rolling without slipping, the center’s speed satisfies v_center = Rω.

This relationship is used in vehicle wheels, conveyor rollers, gears, and bicycle motion. It fails for a wheel skidding across the ground.

Common mistakes

• Assuming all movement is translation.
• Calling rotation motion only when an object changes location.
• Confusing radians with degrees in s = rθ.
• Assuming all points on a rotating body have equal tangential speed.
• Using v = Rω for a slipping wheel without checking the assumption.

Key terms

• Displacement: change in position
• Angular displacement: change in orientation measured in radians
• Angular velocity: rate of angular displacement
• Angular acceleration: rate of change of angular velocity
• Tangential speed: linear speed along a circular path
• Rolling without slipping: rolling motion where v of center equals Rω

Check your understanding — fully worked answers

1. How many radians are in half a revolution?

   Worked answer: π radians.

2. A disk rotates 6 rad in 2 s. Find average angular velocity.

   Worked answer: ω = Δθ/Δt = 6/2 = 3 rad/s.

3. Find rim speed for r = 0.50 m and ω = 10 rad/s.

   Worked answer: v = rω = 0.50 × 10 = 5.0 m/s.

4. Why is a point on a spinning wheel accelerating at constant angular speed?

   Worked answer: Its velocity direction changes continuously, producing centripetal acceleration.

5. A wheel of radius 0.40 m rolls without slipping at 2 m/s. Find ω.

   Worked answer: ω = v/R = 2/0.40 = 5 rad/s.

Key takeaways