7. (Problem Solving)

# Angular Displacement

The official unit of angular displacement $\Delta\theta$ is in radians, but it can be expressed in several different ways.-
*A wheel turns through 5 radians.*This is pretty straightforward: $\Delta\theta=5\u{rad}$ -
*A wheel spins through 90°.*To convert from degrees to radians, you can multiply by ${\pi\u{rad}\over 180\deg}$: $$\Delta\theta = 90\deg \times {\pi\u{rad}\over 180\deg} = {\pi\over 2}\u{rad}$$ -
*A wheel spins around 5 times.*"Times" is usually a synonym for "revolutions". To convert from revolutions to radians, multiply by ${2\pi\u{rad}\over 1\u{rev}}$: $$\Delta\theta = 5\u{rev}\times {2\pi\u{rad}\over 1\u{rev}} = 10\pi\u{rad}$$

Remember that a counterclockwise rotation means $\Delta\theta>0$, and a clockwise rotation means $\Delta\theta<0$.