Wheel kinematics – find angular velocity from rim speed A wheel has diameter 10 m. If a point on its rim moves with linear speed 50 m/s, what is the angular velocity ω of the wheel?

Introduction / Context:
Relating linear speed at a rim to angular speed is a routine conversion in machinery, conveyors, turbines, and vehicle dynamics. The relation v = r * ω directly connects geometry to rotation rate.

Given Data / Assumptions:

• Diameter D = 10 m → radius r = 5 m.
• Linear speed at the rim v = 50 m/s.
• Pure rolling rotation without slip at the rim point relative to the wheel.

Concept / Approach:

Use v = r * ω, where v is the tangential speed and r the radius to the point of interest. Solve for ω = v / r.

Step-by-Step Solution:

Verification / Alternative check:

Units: v/r = (m/s) / m = 1/s = rad/s, consistent for angular speed. Double the radius at same v would halve ω, matching intuition.

Why Other Options Are Wrong:

20, 15, and 5 rad/s correspond to incorrect r or arithmetic; 25 rad/s would require v = 125 m/s at r = 5 m.

Common Pitfalls:

Using diameter instead of radius in v = r * ω; mixing rpm with rad/s without converting.

Final Answer:

10 rad/s