Weight, apparent weight, and weightlessness

0:00- [Instructor] You probably know that weight has something to do with gravity, right? But did you know that you can be weightless even with gravity and you can feel a lot of weight even without gravity? (laughing) When I first heard about this, I was like, "Wait, what?" (laughing) So in this video, let's understand exactly what weight is, how do we define it in science, and look at all the nuances, all right?

0:22Okay, so what exactly is weight? How do we define it? So let's just write that down. So we think of weight as a gravitational force that's exerted by a massive, think astronomical body, on a relatively small nearby body. Let's take some examples, okay? So if you consider the Earth and an apple nearby, we know that the Earth exerts a gravitational force on the apple.

0:44Now look, this force is exerted by a massive body on a relatively small body nearby. So this qualifies as weight. Similarly, what if there was an apple close to the Moon, for example? Well, again, Moon is a very massive astronomical body compared to the apple.

1:00So the gravitational force that the Moon exerts on the apple also qualifies as weight. Of course, the gravitational force due to the Moon could be quite different compared to that of the Earth, so the weight would be different, but nevertheless, we can call that force as weight.

1:14In contrast, what about the gravitational force that the Earth exerts on the Moon? Can we call that as weight? Well, Earth is more massive compared to the Moon, but it's not that much more massive. I mean, both of these are considered to be massive astronomical bodies, right?

1:30And also they're not very close to each other. They're very far away. This is not to scale. So now it doesn't make sense anymore to think about this force as a weight. We no longer think of this force as a weight. We'll just call that as gravitational force. So the idea of weight only makes sense when you're dealing with a very massive body exerting a gravitational force on a very tiny, relatively very tiny body nearby.

1:49Okay, with that definition, now the question is, can we find an expression for weight? And the answer is yes because weight is just the gravitational force at the end of the day, right? So we know from Newton's universal law of gravity that the gravitational force between any two masses, the magnitude of the gravitational force between any two masses will always, you know, be given by this formula where m1 and m2 are the masses of the interacting bodies and r is the distance between the centers of masses and G is the universal constant.

2:19In our case, let's say the mass of the apple is small m and the mass of the Earth is capital M, and let's consider the distance between the centers to be small r. So if you plug that in, this is what we would get. And now we can rearrange this a little bit because G to capital M divided by r square, that represents the gravitational field that the Earth produces at that particular point.

2:44So we can replace this with G. So we can now write the gravitational force equals the mass of the apple multiplied by the gravitational field at that particular point. And of course, since we already know the directions, we can get rid of the vector signs and we just write F equals mg.

3:00So this is the expression for weight. So if you want to calculate the weight of any object, just take the mass of that object and multiply it with the gravitational field at that point. So now let's try to calculate the weight on the Earth's surface. For that, we should first calculate the gravitational field on the Earth's surface.

3:19So do we know the mass of the planet? Yes. Mass of Earth is 6 times 10 of the power 24 kilograms, but what about r? Well, remember, r is the distance from the center of the Earth to the center of this mass. Now, since Earth is so big and this object is very close to the Earth, we can approximate that to be just the radius of the Earth.

3:38I mean, think about it. The radius of the Earth, this is basically 6,400 kilometers. Now, if this apple was even a couple of kilometers above the Earth's surface, you know, you add a couple of kilometers to 6,400 kilometers. Pretty much the same, right? It doesn't matter much.

3:53So we can approximate r to be just the radius of the Earth, and we know the value of G, the universal constant. So if we make the approximation where r is just the radius of the Earth and we plug all these numbers, we get the value of G to be the famous 9.8. That's the gravitational field close to the Earth's surface, and its units are going to be, remember, it's newtons per kilogram.

4:17We can also approximate it to be 10 newtons per kilogram, and as long as things are not moving too far away from the surface of the Earth, which is pretty much the case in everyday life, we can treat the gravitational field close to Earth to be a constant. And yes, if you're wondering, the acceleration that an object has purely due to gravity if there are no other forces acting on it, that also is 9.8 meters per second square close to Earth.

4:41So that means if you wanna calculate weight of an object close to Earth, we'll just take the mass of an object and multiply it by 9.8. So if mass of this apple was one kilogram, its weight would be 9.8 newtons. Okay, we now know how to calculate the weight, but how do we measure it experimentally?

4:58Well, you might say we measure it using a weighing scale, but let's see if the weighing scale actually measures the force of gravity. So if you were to stand on a weighing scale, what are the different forces acting on you? Well, there is the force of gravity, your weight, but then the weighing scale also pushes up on you, so there is that normal force.

5:16Now the question is, which of the two forces does the weighing scale actually measure, the force of gravity or the normal force? Well, the weighing scale measures the force it's being pressed on. In other words, the weighing scale measures the force that you are exerting on it, which is equal and opposite to the force that the weighing scale is exerting on you.

5:37In other words, what the weighing scale registers is not the force of gravity but the normal force. So the weighing scale measures the normal force, and that's why we also call normal force the apparent weight because that's what it apparently measures. So the question is, how are the two different from each other?

5:54This can be slightly confusing, so let's do it slowly. Let's consider the first case where you're just standing still on Earth. So your acceleration is 0, okay? So in this case, how do they compare? Well, we can use Newton's second law in the vertical axis. Newton's second law says the net force acting equals mass times acceleration.

6:12If you do this intuitively, if the acceleration is 0, net force must be 0. So the two forces must cancel out, meaning the normal force should be exactly equal to the force of gravity in this particular case. But of course, if you do it more systematically, which is applicable in general, then we would consider one direction to be positive.

6:27We'll choose upwards to be positive. Then the net force over here would be FN minus Fg, and that would be equal to m times a, and a equals 0. So our equation would be FN minus Fg equals 0, which gives us the same thing. The normal force should equal to the force of gravity, which is exactly equal to mg.

6:47So in this particular case, the weighing scale registers exactly the same thing as your gravitational force. So apparent weight equals the actual weight. But what if you are not standing still? What if you are accelerating upwards like in an elevator? Now do you think the apparent weight or the normal force would be the same as the gravitational force or would it be more or would it be less?

7:11Why don't you pause the video and think about this? Again, let's first do this intuitively. Now there's a net force upwards because the elevator is accelerating upwards. If the normal force was equal to the force of gravity, they would've canceled out, but if you want a net upward force, this time, the normal force must be larger than the force of gravity.

7:30So we would expect the apparent weight to be larger than your real weight. Again, we can try and do this systematically now. Since the upward direction is positive, this time, FN minus Fg would equal m times a, and therefore the normal force equals, look, the force of gravity plus ma.

7:50The normal force would be larger than the force of gravity. Your apparent weight would be larger than your real weight. Your weighing scale now would register a bigger value for your weight because your apparent weight is larger. In fact, you would feel heavier.

8:06So look, apparent weight need not always be the same as your force of gravity, not be the same as your real weight. Okay, let's consider now the opposite case. What if your elevator was accelerating down this time? What would happen? Well, this time, since the net force is downwards, we would expect the normal force to be smaller than the force of gravity.

8:29Again, if we were to do this systematically, this time, we would get FN minus Fg equals minus ma because the elevator is accelerating down. Downwards is negative, and so look, normal force will be smaller than the force of gravity. Your apparent weight this time is smaller than your actual weight.

8:45Your weighing scale registers a smaller weight. You feel a smaller weight over here. But finally, the most interesting case is what if your elevator is in free fall? Okay, then it's accelerating down at g. It's the same case where the acceleration is g. Well, now I have the same equation because it's the same case except that a will be equal to g.

9:10But hey, F is the same thing. The force of gravity is mg. So this cancels out, giving you the normal force to be 0. In other words, when you are freely falling, your apparent weight is 0. There is no contact force anymore. In fact, you will feel weightless. So if you are from the perspective of the falling, you know, elevator, it will feel as if you're just floating over there.

9:35So look, even though there is a force of gravity acting on you, you feel weightless. Your apparent weight is 0, but this number would make us very curious. You see, if you're freely falling, you feel weightless as if there is no force of gravity. Even though there is, it feels like there is no force of gravity.

9:53It feels like there is no weight. Your apparent weight is 0, which is very similar to what would happen if you were in deep intergalactic space where there wasn't any gravity. Imagine you're far away from any masses or almost negligible gravity. So there is no force of gravity.

10:06There is no net force, no acceleration. So even in this case, normal force would be 0. So even in this case, your apparent weight is 0. So in both cases, your apparent weight is 0, which means from inside the elevator, in both cases, you will feel pretty much the same like you are floating.

10:23But of course, here there is gravity. You are freely falling. But over here, there is no gravity and you're just floating. But the question is, could you tell the difference from inside the elevator without telling, without looking outside? Is there any experiment that you can do that tells you whether you're here or here?

10:41The answer seems to be no. Any experiment you do will just tell you that the normal force is 0, that your apparent weight is 0, but it can't tell you whether you are in free fall near Earth, or any planet for that matter, or you're just floating far away in deep intergalactic space.

10:56The two situations seem very equivalent to each other. Now what's more interesting is what if you had to attach a couple of rockets over here so that the whole elevator is accelerating up but in deep intergalactic space? Now there would be a net force on you because you're accelerating up along with the elevator, but the only force acting on you is the normal force.

11:17So the normal force, which is the net force, must equal mg, which means you will measure an apparent weight over here, which is pretty much the same as you would measure standing just still on Earth, but the situation is very different. Here, there is no gravity.

11:34Still you can measure and you'll feel the weight mainly because of your inertia because you are accelerating. It's your inertia that gives you the feeling of weight that, you know, because of weight, you'll measure an apparent weight. So again, could you do an experiment that tells you whether you are in this situation or this?

11:51The answer seems to be no. Again, these two seem to be equivalent, which means there's absolutely no experiment that you can do inside an elevator that tells you whether your elevator is accelerating at 9.8 meters per second square in deep intergalactic space where there's no gravity or you're just standing on Earth.

12:08There's no experiment that can distinguish between them. So we call this the equivalence principle. It was this thought experiment that planted the seeds for what would eventually become general relativity. It's an upgraded model of gravity which made predictions and explained observations that Newton's model couldn't.