Now we're going to talk about why sometimes physics is cruel. So in this case, it's actually related to power. Physics is cruel a lot of time. So if I lift this weight and I sit here and I'm feeling the burn and everything, I'm definitely doing work when I lift the weight. I'm applying a force up and I'm getting a displacement up. But when I just hold the weight out like this, which I can barely do, what would physics say? I'm not doing any work, right? Because there is no displacement. Oh my God, I'm definitely doing work and physics says I'm not. That's a case where physics is cruel. Physics would say there's no work because there was no displacement, I was just holding it still. In that case, the reality has to do with internal energy. I am not a solid object like this table. I'm a soft, squishy mass and my body has to do work to make my arm rigid enough to hold up the weight. So work is happening inside, I'm doing internal work in here, lots of workload, little myosins walking along, but there's no work being done on the weight. So physics is more specific, it wouldn't seem so cruel. You're not doing any work on the weight when you hold it like that. Let's look at another case where we can actually run some numbers or some equations about work. So another case where physics is cruel is, say you are loading a truck with all your stuff because the semester is over almost and you've got to take all your stuff home. You've got a load in the back and you need to build a ramp, you need a ramp, and you're going to put your stuff on a cart, roll it up. You've got to apply a certain force to overcome the friction and roll it up. You say, I could get a really short ramp like this. That would be hard, right? Here's all your stuff like that, right? All these heavy books that you've been studying all semester and you push it up here and into the truck. That'd be one option. Or you could say actually, what would probably be a lot easier is I've got a really long ramp like this. So instead, what you could do is you could put all your heavy books that you've been studying all semester on this long ramp and push them all the way up the long ramp. You would say this was a lot less work, right? A lot easier to do this than this. Of course, what's physics going to say? Physics is going to say, you did the same amount of work. Because the work you did is the potential energy you gained, in both cases, it's the mass of all your books times gravity times how much you raised it, delta y. That's the work you did as you went from here to here, there's delta y, and here to here is the same delta y. You say no, that's not right, because this was a lot more work than this. But you don't mean work, you mean power. So let's compare the power when we go at a steep angle compared to when we go at a shallow angle. Let's see how much power it takes as a function of angle. So what we're going to say is in both cases, push the cart at a steady velocity v, right? We're going to say, what is the power? Now let's see. It's the amount of work you do. It's the change in work, delta w, over delta t. So it's this mg delta y over delta t. Just what physics said we did, mg delta y. But we're going at a constant velocity, so then what do we call delta t? Well, let's see. I guess it takes longer here because we're going farther, right? So if we say the ramp's length is d, in both cases it depends on the angle of course, then dv equals vt, then in the bottom you'd put d over v. So mg delta y, and this must be d over v, like that. Okay. We could do that. But then let's see, d is going to depend on the angle. So we got to figure that part out. So we've got to say, well, doesn't really matter, we're getting it as a function of theta. Sine theta is opposite over hypotenuse. Sine theta is opposite over hypotenuse d. So d is delta y over sine theta. So this is mg delta y over delta y over sine theta. I guess I could go ahead and put the v up here as well. Okay. So now we see this is interesting. The delta ys canceled and we get that it is mgv sine theta. So what you find is the work is the same for the two angles of the ramp, but the power is different. Right? When the angle is really small, sine of theta is small, small angle, very little power. When the angle is really big, sine of theta gets bigger, got a higher angle, you got to put out more power to push it at the same speed. So when you say it feels like more work, you really are thinking of power. How much the rate at which your body is having to expend energy sometimes is more important than actually how much energy you expend. More like what you experience. You may look at this and see, actually, we could have calculated this a bunch of ways. You say, wait a minute, this looks like f and v, almost like that dot-product, but it's sine instead of cosine. That just has to do with which way I define the angle, right? If we are to find the angle up here, it would have come out cosine, then you could just call this f dot v. So you can calculate in a lot of different ways, but you'll always get the same thing. That in this case, the power depends on the angle so we feel easier when we go at a low angle. So physics is cruel, it is definitely true. All these little cases come up where it tells you you're not really doing what you think you're doing. We're going to see more of them as we keep going with energy.