Physics in Action: A Downhill Race

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Have you ever wondered why it is that some fancy sports cars have fancy magnesium wheels, what a ballerina's doing when she pulls in her arms to execute a pirouette? Both of these questions have answers relating to the concept of moment of inertia.
Now, moment of inertia is a formidable-sounding phrase, but it's simply a property of an object that measures its resistance to being rotated. Just as an object has a mass, which is a quantitative measure of how difficult it is to accelerate it linearly, like this or like this, it also has a property called moment of inertia, which measures how hard it is to get it accelerated as a rotation, in this case, the rotation about the center of mass. Now, for an object that's sort of complicated, like this, the moment of inertia is some complicated formula. But suppose the object were just a hoop, like this. The moment of inertia would be a very simple formula. The moment of inertia of this hoop would simply be the mass of the hoop times the square of its radius. For some object, like a sphere, it's 2/5 times the mass of this sphere times the square of its radius and so on. For every object, there's a simple formula like that.
Now, what's the significance of moment of inertia? Moment of inertia, as I said, measures the resistance of an object to being accelerated as rotational acceleration. Now, what's interesting is that two objects, like these two that I'm holding over here, which have the same mass and the same radii, can have very different moments of inertia, because of the difference in the mass distribution. Note that, in this object, most of the mass, which is in these brass objects, is located at the perimeter of the hoop. By contrast, in this object, most of the mass, the same mass, is located near the center of the hoop. It turns out that an object where most of the mass is close to the center has a smaller moment of inertia than the same object with the same mass and the same radius with the mass is distributed farther out. Now, as a consequence of this, this object will offer much less resistance to being rotated than will this object, and we can demonstrate that in the following way.
Suppose we carryout a race, where we let these two hoops roll downhill, starting from the same point? Now, if what I said is right, the object that has the smaller moment of inertia, which is the object closer to you over here, should win the race, because it is easier to get this object to start rotating than it is for the other object. So here we have them lined up at the starting line. Like the Kentucky Derby, I'm going to pull away this black thing and let's see how the race goes.
Indeed, this object with the smaller moment of inertia won the race, precisely because the moment of inertia measures the ease with which an object can start rotating. And, of course, that's what has to happen if these objects are going to undergo a race. The object with the small moment of inertia won the race.
Now, let's take a look at these two objects. Both of these are cylinders, but this is obviously a lot more massive that this. This is a wood cylinder, a thin wood cylinder, this is a thick aluminum cylinder. Let's weigh them to show you how different they are in mass. Well, this is a mass of a little more than 3 kilograms and this is a mass of about 400 or 500 grams. They clearly have very different moments of inertia. This is a much bigger moment of inertia than this. Now, if you listened to what I said before, you would think that it would be much more difficult to get this object rolling than this, than the wood object, and let's see if that's really true by conducting yet another race. I'm going to line them up again. This is the big object with the moment of inertia and this is the small object. Let's line them up this way.
They ended up crossing the finish line at exactly the same time. Now, what was wrong with what I said before? I just got through explaining to you that the object with the bigger moment of inertia is much more difficult to accelerate than the object with the small moment of inertia. Yet these two objects with very different masses and very different moments of inertia, in fact, tied in the race. What's going on over here?
Well, in a special case where the force causing these objects to accelerate is gravity, we have a special circumstance, and that's given by this formula over here. Velocity, using conservation of energy, you can show that the final velocity of these objects is simply given by the square root of the number 4/3 times g, the acceleration of gravity, times the vertical height for which they fell. Notice, this is a key point, that the formula does not depend in any way on the mass of the object, provided both of these objects are cylinders, namely, they have the same mass distribution. That's the reason why, when the acceleration is caused by gravity, they can, in fact, accelerate down the ramp at exactly the same rate.
What would happen if we take three very different objects, this wooden disk, this ball and this hoop, and we allow them all to engage in a race down the same ramp? What's going to happen in this situation? Well, you know enough to think about this and to come up with the right answer, but let's give you a little bit of help. Let's start out by looking at what's going on. We have three objects, represented by this circle, accelerating down a plane through a vertical height, h. After they finished traveling through a vertical height, h, the initial potential energy, mgh, has been converted into kinetic energy by this formula over here. mv^2 represents the contribution to the kinetic energy from the motion of the center of mass of each object and I^2, where I is the moment of inertia, represents the additional kinetic energy, simply because of rotation about the center of mass. And the sum of these two is, in fact, always equal to mgh, the difference in potential energy between the beginning and the end.
Well if we solve this equation for what the final velocity is, we see that, in fact, the final velocity depends in some way on I, on the moment of inertia. So, in general, objects with different moments of inertia will achieve different final velocities and one or the other is going to win this race. If we put in the actual values for the moment of inertia this sphere, of this hoop and of this disk, we're going to find that the final velocity will be different in these three cases. For the hoop the velocity is the square root of 1 times gh, where g is the acceleration of gravity and h is again the vertical height to which the ball fell. For the cylinder, the velocity is the square root of 4/3 times gh, that 4/3 arising from the different matter distribution in the cylinder. And, finally, in a sphere, the velocity is the square root of 10/7 times gh. Notice the coefficients are 1, 4/3 and 10/7. They're all different. And because they have different coefficients over here, we expect them to have different velocities. That's what this formula says. So if we were to carryout this race, we expect there will be no ties. The one with the biggest coefficient is going to win. Let's see what happens if we actually do the demonstration. Ready, set go.
The Physics of Extended Objects
Rolling
Physics in Action: A Downhill Race Page [2 of 2]