Notes on Relativity and Cosmology - Physics Department, UCSB
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Notes on Relativity and Cosmology - Physics Department, UCSB

Notes on Relativity and Cosmology - Physics Department, UCSB Notes on Relativity and Cosmology - Physics Department, UCSB

web.physics.ucsb.edu
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27.09.2014 Views

150 CHAPTER 6. DYNAMICS: ENERGY AND ... Before After After the experiment, it is clear that the box has moved, and in fact that every single atom in the box has slid to the left. So, the center of mass seems to have moved! But, Einstein asked, might something else have changed during the experiment which we need to take into account? Is the box after the experiment really identical to the one before the experiment began? The answer is: “not quite.” Before, the experiment, the battery that powers the laser is fully charged. After the experiment, the battery is not fully charged. What happened to the associated energy? It traveled across the box as a pulse of light. It was then absorbed by the right wall, causing the wall to become hot. The net result is that energy has been transported from one end of the box (where it was battery energy) to the other (where it became heat). Battery fully charged Before Battery not fully charged After Hot wall So, Einstein said, “perhaps we should think about something like the center of energy as opposed to the center of mass.” But, of course, the mass must also contribute to the center of energy... so is mass a form of energy? Anyway, the relevant question here is “Suppose we want to calculate the center of mass/energy. Just how much mass is a given amount of energy worth?” Or, said another way, how much energy is a given amount of mass worth? Well, from Maxwell’s equations, Einstein could figure out the energy transported. He could also figure out the pressure exerted on the box so that he knew how far all of the atoms would slide. Assuming that the center of mass-

6.3. ON TO RELATIVITY 151 energy did not move, this allowed him to figure out how much energy the mass of the box was in fact worth. The computation is a bit complicated, so we won’t do it here 6 . However, the result is that an object of mass m which is at rest is worth the energy: E = mc 2 (6.3) Note that, since c 2 = 9 × 10 16 m 2 /s 2 is a big number, a small mass is worth a lot of energy. Or, a ‘reasonable amount’ of energy is in fact worth very little mass. This is why the contribution of the energy to the ‘center of mass-energy’ had not been noticed in pre-Einstein experiments. Let’s look at a few. We buy electricity in ‘kilowatt-hours’ (kWh) – roughly the amount of energy it takes to run a house for an hour. The mass equivalent of 1 kilowatt-hour is m = 1kWh c 2 = 1kWh c 2 In other words, not much. 3600sec 1000W hr. 1kW = 3.6 × 106 9 × 10 16 = 4 × 10−10 kg. (6.4) By the way, one might ask whether the fact that both mass and energy contribute to the ‘center of mass-energy’ really means that mass and energy are convertible into one another. Let’s think about what this really means. We have a fair idea of what energy is, but what is mass? We have not really talked about this yet in this course, but what Newtonian physicists meant by mass might be better known as ‘inertia.’ In other words, mass is defined through its presence in the formula F = ma which tells us that the mass is what governs how difficult an object is to accelerate. 6.3.4 Mass, Energy, and Inertia So, then, what we really want to know is whether adding energy to an object increases its inertia. That is, is it harder to move a hot wall than a cold wall? To get some perspective on this, recall that one way to add energy to an object is to speed it up. But we have already seen that rapidly moving objects are indeed hard to accelerate (e.g., a uniformly accelerating object never accelerates past the speed of light). But, this just means that you make the various atoms speed up and move around very fast in random ways. So, this example is really a lot like our uniformly accelerating rocket. In fact, there is no question about the answer. We saw that heat enters into the calculation of the center of mass. So, let’s think back to the example of you walking in a canoe floating in water. If the canoe is hot, we have seen that it counts more in figuring the center of mass than when it is cold. It acts like a heavier canoe and will not move as far. Why did it not move as far when you walked in it in the same way? It must have been harder to push; i.e., it had more inertia when it was hot. Thus we conclude that adding energy to a system (say, charging a battery) does in fact give it more inertia; i.e, more mass. 6 Perhaps some senior physics major would like to take it up as a course project?

150 CHAPTER 6. DYNAMICS: ENERGY AND ...<br />

Before<br />

After<br />

After the experiment, it is clear that the box has moved, <strong>and</strong> in fact that every<br />

single atom in the box has slid to the left. So, the center of mass seems to have<br />

moved! But, Einstein asked, might something else have changed during the<br />

experiment which we need to take into account? Is the box after the experiment<br />

really identical to the <strong>on</strong>e before the experiment began?<br />

The answer is: “not quite.” Before, the experiment, the battery that powers<br />

the laser is fully charged. After the experiment, the battery is not fully charged.<br />

What happened to the associated energy? It traveled across the box as a pulse<br />

of light. It was then absorbed by the right wall, causing the wall to become<br />

hot. The net result is that energy has been transported from <strong>on</strong>e end of the box<br />

(where it was battery energy) to the other (where it became heat).<br />

Battery<br />

fully charged<br />

Before<br />

Battery not<br />

fully charged<br />

After<br />

Hot wall<br />

So, Einstein said, “perhaps we should think about something like the center of<br />

energy as opposed to the center of mass.” But, of course, the mass must also<br />

c<strong>on</strong>tribute to the center of energy... so is mass a form of energy?<br />

Anyway, the relevant questi<strong>on</strong> here is “Suppose we want to calculate the center<br />

of mass/energy. Just how much mass is a given amount of energy worth?” Or,<br />

said another way, how much energy is a given amount of mass worth?<br />

Well, from Maxwell’s equati<strong>on</strong>s, Einstein could figure out the energy transported.<br />

He could also figure out the pressure exerted <strong>on</strong> the box so that he<br />

knew how far all of the atoms would slide. Assuming that the center of mass-

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