The Physics Philes, lesson 41: Avenging Angular Acceleration

In which equations are derived, Blu-rays spin, and Hulk movies are lamented.

The last couple of weeks we’ve been discussing angular velocity and acceleration. You may have gotten a case of déjà vu; calculating angular velocity and acceleration is very similar to calculating straight line motion. Well get ready for an even déjà vu-ier post, because now I’m going to talk about rotation with a constant angular acceleration.

Basically, don’t let the rotation confuse you. If you understand straight line motion of an object with constant acceleration, you can understand this. So let’s dive in!

First, let’s get the equations. We know that the average angular acceleration is the change in angular velocity divided by the change in time. If the time interval is 0 to some time in the future t, that equation would look like this:

Let’s get rid of that ugly fraction:

And ta-da! A handy little equation for problems with constant angular acceleration. But we’re not done! We know from the last couple of weeks that the the average angular velocity is is the total angular displacement divided by the time. And with constant angular acceleration, the angular velocity changes at a uniform rate. So to find the average angular velocity between times 0 and t, we divide the sum of the initial and final angular velocities by two. When you mesh those two equations together with the dark wizardry of mathematics, you get:

So there’s another tool for your tool belt. But wait! We’re still not done! Say you need to know the relationship between the angle in question and the time t, but you don’t want to deal the final angular velocity. What do you do? Luckily, we can derive an equation for that very situation.

Remember waaaaay at the top of this post we discovered that the final angular velocity is the sum of the initial angular velocity plus the product of the angular acceleration and time? All we have to do is substitute the left hand side of that equation into the equation immediately above to get rid of that nasty ωz:

Oh gross brackets! Get out of here, brackets!

Ahhhh…much better.

You may be noticing a theme at this point. We’re basically replacing v and x from the constant acceleration formulas with ω and Θ. So you may have already guessed what the equation is for the relationship between θ and the final angular velocity without reference to time t:

Remember, all of these equations are only good for when you’re dealing with an object that has constant angular acceleration. If you use them when the acceleration isn’t constant, you’re going to goof it up.

Hey, I think I even have time today to do an example problem!

You just finished watching The Avengers Blu-ray edition for the 53rd tie and you’re thinking about what a damn shame it is that Hollywood can’t produce a decent Hulk movie. (Or is that just me?) All the while, the Blu-ray is slowing down. The angular velocity of the disc at t = 0 is 27.5 rad/s and it’s angular acceleration is a constant -10.0 rad/s/s. A line PQ that you for no good reason drew on the otherwise pristine Blu-ray lies along the positive x-axis at t = 0. What is the disc’s angular velocity at t = 0.300 s, and what angle does the line PQ make with the positive x-axis at t = 0.300 s?

Let’s draw a diagram!

We have two target variables: the angular velocity at 0.300 second and the the angle at that same time. What do we know? We know the initial angular velocity, the initial angle, the angular acceleration, and the time.

To find the angular velocity, we use the first equation we derived today. All we need to do is plug in the values:

The angular velocity at 0.300 s is 24.5 rad/s! Awesome! Now let’s find the second target variable: the angle line PQ makes with the positive x-axis. For this question, all we need is the second to last equation we derived today:

Hm…one revolution is 2Π rad. So what the heck is 7.80 rad?

So in 0.300 s the line PQ goes around about one and a quarter times Wowza.

Now you know everything about rotation with constant angular acceleration. Well, probably not. But it’s something.

Let me know if I got anything wrong in this post. At one point I got confused between the symbol for angular velocity and acceleration. I think I fixed all of it, but it could get pretty confusing if I didn’t. Please let me know if I goofed.