Kinematics – Linear (Tangential) Speed in Circular Motion A body rotates with angular velocity ω (rad/s) along a circular path of radius r (m). What is its linear speed v?

Introduction / Context: Uniform circular motion links angular quantities (radians, rad/s) with linear kinematics (meters, m/s). Converting between angular velocity and linear speed is fundamental in rotating machinery, gears, and belt drives.

Given Data / Assumptions:

• Angular velocity ω in rad/s.
• Radius r in meters.
• Body moves along a circular path without slipping.

Concept / Approach: Arc length s traveled in time t is s = r * θ. Differentiating with respect to time gives linear speed v = ds/dt = r * dθ/dt = r * ω.

Step-by-Step Solution: Start from s = r * θ. Differentiate: v = ds/dt = r * dθ/dt. Recognize dθ/dt = ω. Thus v = r * ω.

Verification / Alternative check: Units: ω (rad/s) is dimensionally 1/s; multiply by r (m) to get m/s, which is correct for linear speed.

Why Other Options Are Wrong: ω / r and r / ω invert relationships incorrectly (wrong units). ω^2 / r and ω^2 * r are associated with centripetal acceleration (a_c = ω^2 r), not speed.

Common Pitfalls: Confusing v = ω r with a_c = ω^2 r; remember speed vs acceleration.

Final Answer: ω * r