8. (Problem Solving)
Angular Velocity
The official unit of angular velocity $\omega$ is in radians/second, but it can be expressed in several different ways.- A wheel spins at 5rad/s. This is pretty straightforward: $\omega=5\u{rad/s}$
- A wheel spins around 5 times per second. We can think of this in two different ways:
- This is the angular velocity measured in revolutions per second, so we can convert it to rad/s by multiplying by $2\pi\u{rad}\over 1\u{rev}$: $$\omega = {5\u{rev}\over s} \times {2\pi\u{rad}\over 1\u{rev}} = 10\pi\u{rad/s}$$
- "Turns per second" is sometimes referred to as the frequency $f$. When we're given the frequency then $\omega = 2\pi f$: $$\omega = 2\pi(5\u{rev/s}) = 10\pi\u{rad/s}$$
- A wheel takes 4 seconds to turn around once. This is referred to as the period $T$ of the wheel's rotation, and is given in units of s/rev. If we flip it upside-down, we get the frequency: $$f=\frac1T = \frac1{4\u{s/rev}} = \frac14\u{rev/s}$$ We can then convert it to radians as above: $$\omega = 2\pi f = \frac14\u{rev\over s} \times {2\pi\u{rad}\over 1\u{rev}} = {2\pi\over 4}\u{rad/s}$$ Or more generally, $$\omega = {2\pi\over T}$$ This is probably the trickiest thing in these problems, because it looks like you're being given $\Delta t$. Remember that $\Delta t$ is a duration, the time between some initial and final event, the time during which the wheel is speeding up or slowing down.