# So what is happening in this angular kinematics equation?

-The wheel of a car has a radius of 20 cm. It initially rotates at 120 rpm. In the next minute it makes 90 revolutions. What is the angular acceleration?_

So the answer is solved by using one of the angular kinematic equations. More specifically delta theta. The problem I am having trouble understanding is the answer which probably stems from my poor fundamentals.

The answer is: (90x2pi) = 4pi(60) + 1/2 (alpha) (60)^2

So where does 90x2pi come from?

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I'm assuming by "in the next minute" you really mean something like, "after a steady acceleration for one minute it reaches 90 rpm" as otherwise your answer isn't right (and the problem becomes a lot more sophisticated). – Carl Brannen Jul 21 '11 at 8:01

The equations you're using are equations for $\theta$, the angle through which the wheel rotates. In physics, we measure angles in radians. There are $2\pi$ radians in a circle, so $90$ revolutions is $90*2pi$ radians.

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Mixing units is always a recipe for trouble.

As you know, angular acceleration follows similar laws as linear acceleration. Specifically, the angle $\theta$ at time $t$ is given by

$$\theta(t) = \theta(0) + \omega(0) t + \frac12\alpha t^2\tag1$$

which is the analog of

$$x(t) = x(0) + v_0 t + \frac12 a t^2$$

In your case, the angular velocity is expressed in rpm - revolutions per minute - and we usually want to convert to radians per second. $2\pi$ radians in one revolution, 60 seconds in a minute. So

\begin{align}\theta(0) &= 0\\ \omega(0) &= 120 * 2 \pi / 60\\ &= 4\pi\\ \theta(t) &= 90\ rev\\ &= 90 * 2 \pi\\ &= 180\pi\\ t &= 60 s\end{align}

Solving $(1)$ for $\alpha$:

\begin{align} \alpha &= 2\frac{\theta(t) - \omega(0) t}{t^2}\\ &=2\frac{180\pi - 240\pi}{3600}\\ &=-\frac{\pi}{30} \frac{rad}{s^2}\end{align}

You could keep the entire thing in revolutions and minutes - then the equation (and the numbers) would be simpler:

$$\alpha = 2\frac{90-120}{1} = -60 \frac{rpm} {min^2}$$

but now you have to get your head around the units of $\frac{rpm}{min^2}$. Once you do, you realize that the speed of the wheel halved - at the end of the minute it's spinning at just 60 rpm (120 - 60 = 60). The fact that the wheel turned 90 revolutions in that minute makes sense now - with linear acceleration, the average speed (90 rpm) is the mid point between the start (120 rpm) and end (60 rpm) speeds...

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