Angular position and velocity

Radians make angular rate a linear rate

An angle in radians is arc length divided by radius, so it connects rotation to ordinary distance without a conversion constant. Angular position θ describes orientation; angular velocity ω describes how fast that orientation changes.

Differentiate arc length

For a point at radius r, arc length s = rθ when θ is in radians. Differentiating gives v_t = ds/dt = r dθ/dt = rω. Differentiating again gives tangential acceleration a_t = rα, where α = dω/dt. The radius is the geometric lever that converts angular motion to linear motion.

An impeller at r = 0.08 m spins at 1800 rpm. Convert to ω = 1800(2π)/60 = 188.5 rad/s; blade-tip speed is v = rω = 15.1 m/s. Using degrees without conversion would corrupt the speed by a factor of 57.3.

Angles wrap but angular displacement accumulates

Orientation repeats every 2π, but a shaft that makes three turns has angular displacement 6π rad. For oscillation or work, use the accumulated angle rather than the wrapped display angle.

Work it: A wheel of radius 0.30 m accelerates from 4 to 10 rad/s in 3 s. Find α, tip-speed change, and angular displacement assuming constant α.