Let's see why work done by gravity is path independent. Created by Mahesh Shenoy.
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- Why does gravity matter the most?(1 vote)
- [Instructor] Suppose I drop two identical balls from the same height. One I just let go of it, so it just falls straight down. Another one, let's say I have a slide and I make it slide down, it rolls down. The question you want to try and answer is in which of the two cases gravity does more work. Okay? So let's see, how to we answer this question? First we need to know how to calculate work. We know work done equals force acting on a body multiplied by the displacement of that body, right? So let's try and figure out the work done in both these cases. Now which is the force that's acting on them? Since we're introducing gravity, that's the force we should talk about, and the gravitational force on both of them must be the same because they are identical balls, they have the same mass, and you might know that gravitational force is mg, so that part must be the same. Now let's look at the displacement part. In the first case, the ball falls straight down. So the displacement is just the height of the ball, and so in the first case, the work done becomes mg times h. But what about the second case? Now in the second case, the displacement is from here to here which is more than the height of the ball, right? This length, as you can see, is larger than this length, larger than the height. And so if we call this as, let's say s, we might say in this case the work done will be mg into this displacement, and therefore we might conclude that more work is done by gravity in this case, right? Well, an important thing to see over here is that the force and the displacement in this case are not in the same direction. You see, they're in a different direction. And whenever we say we are calculating work as force into displacement, that displacement has to be calculated in the direction of the force. And so I guess the goal of the video is to try and figure out whenever the force and displacement are not in the same direction, how do we calculate work done? What is the meaning of this statement? So let's look at these two cases in a little bit more detail, okay. So in the first case, the force of gravity is down and the object just falls down. No surprise. In the second case, gravity's down, but the ball is falling down at an angle. It's sliding down at an angle. So again, not surprising, but what does it mean? This means gravity can make things accelerate in other directions as well, not just down, but in other directions as well, right? But there's a difference between these two. In which case do you think the ball accelerates more quickly? Well, I'm pretty sure you'll agree it's in the first case. In the first case the ball would accelerate very quickly compared to the second one. Here, let me show you this. So if I drop the ball, see how quickly it just fell down. In the second case, I'll make it slide, and now notice it's a little slower than before. Again, to see it clearly, let's do a comparison side by side in slow motion, and you can see immediately the ball here accelerates down, but over here notice it's so much slower. So the ball accelerates much slower here compared to this. So what can we conclude from this? Because the acceleration is slower, we could say the effect of gravity felt by this ball must be less, right, because it's feeling less force and therefore smaller acceleration. So we can conclude that gravity has an effect in this direction, but that effect is smaller compared to the previous one, right? So at an angle, the effect of gravity reduces. So we can say less effect along this angle. Okay, in a third case what we'll do is we'll try to make that object slide horizontally due to gravity. What'll happen? Well, gravity's acting down, and we're trying to make this object go forward, let's say. Do you think this is going to happen? I'm pretty sure you'll agree nothing's going to happen in this case, right? This object is not going to move at all. Again, I can just show you that. I'll just keep an object, this time it's horizontal. Here the ball is moving because my hand is shaking, but I'm pretty sure you'll agree that the ball is just going to be at rest if this was perfectly leveled, right? So in this case, gravity cannot affect the motion at all. So we can say that gravity has no effect in the horizontal direction. Does that make sense? Because it's not able to accelerate this ball. So in this case we can say gravity has zero effect. Now before we put all of this together and try to make sense of this, let me tell you that this is not just for gravity. This can be seen in any force. To give you an example, let's say you take a mop and you push it forward, and you'll expect the mop to accelerate forward and speed up in the forward direction. No surprise, right? But take that mop and now push it at an angle, and again, that force can accelerate it, meaning even though this force is acting at an angle, it has an effect in this direction just like what we saw over here. So this force also has effects in other direction, but again, you might agree that over here the effect is a little smaller, because now the mop will not accelerate as much as it accelerated in this case, right? So the effect has reduced, and eventually, if you push the mop straight down, now you'll see it has absolutely no effect in the horizontal direction. So this force has zero effect in this direction. So if you put it all together, what do we understand? First of all we see that forces can have effects in other directions as well. But what's important is that as the angle between the force and the motion increases, as this angle starts increasing, the effect starts reducing. You see the effect is maximum when the force and the motion are in line with each other. But as the motion starts making an angle with the force, the effect reduces. And when the angle becomes 90 degrees, as you can see, the effect becomes zero. This is the most important thing, okay? This means that forces have no effect in perpendicular direction. It has zero effects in perpendicular direction. So what can we say about the work done in this particular case? Well, let's take an example. So imagine I take a ball and I keep it on a perfectly horizontal slide, and let's think about the work done by gravity. Gravity's pulling it down. Let's say I push this ball now and make it displace horizontally. What is the work done by gravity? Can we just say it is the gravitational force multiplied by the displacement? No, because the angle between the gravity and the displacement is 90 degrees, we can now say gravity has no effect in this direction like we saw earlier. So gravity's not causing this displacement at all, and therefore we can say the work done by gravity is zero, because gravity's not the one that displaced this body. Does that make sense? In a similar manner, imagine somebody's carrying a luggage on their head, and let's say they move forward, displacing that luggage. What is the work done by their force on the luggage? Again, that's zero. Why? Because they're pushing on the luggage up, and this is perpendicular to direction of the displacement. And so we can say this force has no effect on this direction, it did not cause this displacement at all, and therefore the work done by this force on the luggage must be zero. And so, in general, if the force and displacement are perpendicular to each other, work done by that force must be zero. So now let's see how this knowledge helps us in answering our original question. So we want to calculate the work done in this case, right? Now instead of the ball falling, sliding straight down, let's assume, let's assume for a moment that the ball is going on a staircase. So let's assume that the ball is going forward, then coming down, then going forward, then coming down, and so on and so forth. Let's assume it's going that way. I know it's not really doing that, but let's calculate the work in this case first. It'll help us to figure out what happens in the work in the sliding case as well. So what is the work during this entire motion? Well again, it's going to be force times displacement. We know the force, which is mg, but what is the displacement? Should we take this entire displacement, should we add up all these displacements? No, because we know now that during the horizontal motion, gravity's not doing any work. So this displacement doesn't come into the picture. During the vertical motion gravity is doing work. Gravity is pulling and make it displace, so this matters. Again, during the horizontal motion, that displacement doesn't matter. Again, during the vertical motion, it matters. So, during the horizontal motion, since the work done is zero, we can just get rid of the displacement, that displacement doesn't come into our formula. So let's get rid of that displacement. And so the only displacement that matters to us is the one that happens in the direction of the force of gravity, right? And that's what we meant earlier. When you're calculating the work done, what matters is the displacement that happens in the direction of the force. Only that displacement should come into the picture. Okay, so what is this total displacement? Well, we just have to add all of them up. Now if we add all of them up, notice what we get. We just get the entire height of the fall from the ground, which means the displacement over here has to be h itself, and therefore the work done in this case is exactly the same as the work done in this case. So even though the ball is going down on a staircase, going forward and going down and so on, the work done by gravity must be the same. Now, of course, you might say well, okay, but that doesn't answer my question. Our original question was not on a staircase, it was going down on a slide, right? How do we answer that? Well, we'll come to that. Now instead of thinking of a big staircase, let's assume we made the step size smaller, so we had a longer staircase. Okay, now imagine again the ball goes forward and comes down and so on and so forth. What'll be the work done? Well again, we don't have to worry about the horizontal displacements because the work done by gravity would be zero. Right, in the perpendicular direction, gravity has no effect. So the only displacement that matters to us is in the vertical direction, in the direction of the force. Again, if we add up all those displacements, notice we end up with just h. So this means even if I make this staircase smaller, the work done by gravity should still be the same. So now let me make the staircase even smaller. The work done should still be the same. Let's make the staircase even smaller. The work done would still be the same. Now you can imagine if we go on making the staircase smaller and smaller and smaller and smaller, the motion of the ball would start becoming smoother and smoother, and eventually, if you imagine the staircase while it's super, super, super, super, super small, we can pretty much assume that that staircase is now like the slide. And we can now say that the work done in this slide must also be the same. So in both the cases, gravity does exactly the same work. Amazing, isn't it? Now we can extend this even further. Instead of going on a slide, let's say the ball went on some curve like this. What is the work done by gravity in this case? Again, it should be the same. Why? Because again you can think of it as going in a very tiny staircase from here to here, and when you add up all the displacements, only the vertical displacements will matter, and again, the total displacement will still be just h. So even in this complicated curvy example, work done by gravity is still the same. And so now we can extend this in general and say even if our ball is going in some crazy, weird path, then the work done by gravity will again be just the same. So it seems like the work done by gravity doesn't depend on what path it takes to go from one point to another. It only depends upon the height that it covers, right? The height is all that matters. It does not depend on the path. And this is important, because tomorrow if we have our object going in some crazy path and we are asked to calculate the work done by gravity, we don't have to worry too much about the path taken. We just say it's the force into the height that it covers from one point to another. And so what did we learn in this video? By looking at some day-to-day life examples, we saw that forces can have effects in other directions as well. But what's important is that forces have maximum effect in the line of the force, and as the angle between the motion and the force increases, its effect starts decreasing. And eventually when the angle between the force and the motion becomes 90 degrees, we see that the force has zero effect. Forces have zero effect in perpendicular direction, and because of this, if the force and the displacement are perpendicular to each other, that work done by that force should be zero because the force had no effect in that direction. And using this, we were able to figure out that when a ball falls through some height, it doesn't matter what path it takes, whether it falls straight down or it goes through a curve, the work done will be the same. It'll be the force of gravity times the height through which it fell down, and one way to argue this is instead of imaging that it fell through a curve like this, we can imagine it went through a very, very, very tiny staircase, and then when the ball goes forward, the work done would be zero so those displacements don't matter, and the only displacement that'll come into our equation is the vertical displacement, and when you add all of them up, we saw it'll just be the height. And so regardless of what path the ball takes, the work done by gravity doesn't depend on the path, it only depends upon the height through which it falls.