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Dynamics

Rotational Motion Is Described by Angular Position, Velocity, and Acceleration

Angular position θ, ω=dθ/dt, α=dω/dt, constant α (ω=ω₀+αt, θ=ω₀t+½αt²), link v=rω, at=rα, an=rω²

When a rigid body turns about an axis, every point shares the same angle θ. The x, v, a of straight-line motion carry over to rotation as angular position θ, angular velocity ω, and angular acceleration α, and one factor of the radius r links them back to linear speed (v=rω).

Turn the rigid body by θ. Every point off the axis sweeps the same angle, and a point at radius r traces an arc of length s = rθ (with θ in radians).

Angular velocity ω is how fast the angle changes, ω = dθ/dt. On the θ-t graph ω is the slope — raise ω and the line steepens, turning more in the same time.

Angular acceleration α is how fast ω changes, α = dω/dt. On the ω-t graph α is the slope — α>0 keeps speeding up, α<0 slows and then reverses.

Even at the same ω, points farther out move faster. A point at radius r has tangential speed v = rω — drag ω and the r=2 point is always twice as fast as the r=1 point.

When α is constant, the constant-acceleration formulas carry straight over to rotation. Drag α and the area under the ω-t line is the swept angle Δθ — ω=ω₀+αt, Δθ=ω₀t+½αt², ω²=ω₀²+2αΔθ.

In PracticeTo sum up, rotation is a perfect parallel world to straight-line motion. Swap one word at a time — position x → angular position θ, velocity v=dx/dt → angular velocity ω=dθ/dt, acceleration a=dv/dt → angular acceleration α=dω/dt — and the constant-acceleration formulas carry over too: v=v₀+at → ω=ω₀+αt, x=x₀+v₀t+½at² → θ=θ₀+ω₀t+½αt², v²=v₀²+2aΔx → ω²=ω₀²+2αΔθ. The bridge between the two worlds is the radius r: arc s=rθ, tangential speed v=rω, tangential acceleration at=rα, and the direction-changing centripetal acceleration an=rω²=v²r. That is why the rim of a turntable is faster than the center, and why, for the same angle turned, points farther out have a larger linear speed. In the next lesson we move to general plane motion, where a body turns and translates at once (rolling), solved with the rolling condition v=rω and the instantaneous center of rotation.
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