Angular Velocity

• A wheel spins at 5rad/s. This is pretty straightforward: $\omega =5\mathrm{rad}/\mathrm{s}$
• 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 $\frac{2\pi \mathrm{rad}}{1\mathrm{rev}}$: $$\omega =\frac{5\mathrm{rev}}{s}\times \frac{2\pi \mathrm{rad}}{1\mathrm{rev}}=10\pi \mathrm{rad}/\mathrm{s}$$
• "Turns per second" is sometimes referred to as the frequency $f$. When we're given the frequency then $\omega =2\pi f$: $$\omega =2\pi (5\mathrm{rev}/\mathrm{s})=10\pi \mathrm{rad}/\mathrm{s}$$

• 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: $$f=\frac{1}{T}=\frac{1}{4\mathrm{s}/\mathrm{rev}}=\frac{1}{4}\mathrm{rev}/\mathrm{s}$$ We can then convert it to radians as above: $$\omega =2\pi f=\frac{1}{4}\frac{\mathrm{rev}}{\mathrm{s}}\times \frac{2\pi \mathrm{rad}}{1\mathrm{rev}}=\frac{2\pi }{4}\mathrm{rad}/\mathrm{s}$$ Or more generally, $$\omega =\frac{2\pi }{T}$$ This is probably the trickiest thing in these problems, because it looks like you're being given $\mathrm{\Delta }t$. Remember that $\mathrm{\Delta }t$ is a duration, the time between some initial and final event, the time during which the wheel is speeding up or slowing down.