The Physics Philes, lesson 62: Spin Cycle
I guess I should start out The Physics Philes this week by eating a little crow. (Or maybe, since I’m a vegetarian, I’ll just eat some mycoprotein crow replacement?) I did not write a post last week, and I’m so ashamed! I didn’t even post a video or anything. You see, I started school last week and my parents came to visit the weekend before and…well, enough excuses. I’ll try to make it up to you by trying to explain the conservation of angular momentum today.
We cool? OK. Let’s physics.
We’ve been discussing angular momentum. Think about a spinning ice skater. When his or her arms are held out away from their body, they spin relatively slowly. But, when they bring their arms in toward their body, they speed up. What’s happening here? There is no additional torque to speed the skater up, so how is he or she speeding up? The answer has everything to do with the conservation of angular momentum.
As we know from last week, angular momentum is equal to the the moment of inertia times the angular speed. Think about that spinning skater again. When the skater has their arms splayed out, the moment of inertia is high. When their arms are pulled in, the moment of inertia is low. In order to keep the angular momentum steady, as the skater pulls their arms in the angular speed needs to compensate. So, assuming there is no additional torque on the body, as the skater pulls in his or her arms, the moment of inertia decreases, which means that the angular speed of the skater has to increase.
This, in a nutshell, is conservation of angular momentum. Kinda cool. Now let’s try to do a problem.
Let’s return to our spinning ice skater. Let’s say that the angular speed while their arms are extended is 1.50 rad/s. The skater pulls their arms in and the angular speed increases to 15.0 rad/s. By what factor does the moment of inertia change and by what factor does the rotational kinetic energy change?
There are two questions here we need to answer: the factor by which the moment of inertia changes and the factor by which the rotational kinetic energy change. We’ll need to use two equations to find the answers: the equation for kinetic energy and the equation for angular momentum.
Where L is the angular momentum, I is the moment of inertia, and ω is the angular speed.
Rotational kinetic energy:
Where K_rot is the rotational kinetic energy, I is the moment of inertia, and ω is the angular speed.
Let’s figure out the first question first: by what factor does the moment of inertia change?
Since there is no additional torque in the question, that we can use conservation of angular momentum to solve this problem. We can set the angular momentum after equal to the angular momentum before, like so:
Notice that I just replaced L with Iω. Now we can rearrange the equation to get the ratio of the moment of inertia before and after the change in arm position:
Now we just plug in the numbers…
…and ta-da! We know by what factor the moment of inertia changed. The moment of inertia after the skater’s arms are pulled in is only 1/10th that of the moment of inertia before.
We aren’t done yet. Now we need to figure out by what factor the rotational kinetic energy changed. To do this, we need to calculate the ratio of the skater’s kinetic energy before and after:
Then, just insert the numbers and solve:
The rotational energy the skater has after they pull their arms in is ten times greater than when he or she had their arms out. This is what we’d expect from what we’ve all observed.
Well, I hope I’ve made up for not posting any physics last week. Seriously, I feel terrible about it. But, with any luck, I’ll be back next week with more physics word nourishment.