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# Work done by gravity (path independent)

## Video transcript

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 wanna try an answer is in which of the two cases gravity does more work okay so let's see how do 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 it's acting on them since you're interested in gravity that's the force we should talk about and the gravitational force on both of them must be the same because they're identical balls they have the same mass and you might know the 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 which 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 one 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 is down but the ball is falling down at an angle it's sliding not 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 agree it's in the first case in the first case the ball will accelerate very quickly compared to the second one here let me show you this so if I drop the ball see how quickly just fell down the second case I'll make it slide and now notice it's a little slower than before again to see it clear clearly let's do a comparison side by side it's 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 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 could say less effect along this angle okay the third case what we'll do is we'll make that we'll try to make that object slide horizontally due to gravity what will happen well gravity is acting down and we're trying to make this object go forward let's say do you think this is gonna happen I'm pretty sure you agree nothing's gonna 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 yeah the ball is moving because my hand is shaking but I'm pretty sure you agree that the ball is just gonna 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 is unable 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 an example let's say you take a mop and you push it forward then you'll expect the mop to accelerate forward will 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 notice 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 them up 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 direction 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 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 is pulling it down let's say I push this ball now and make it displaced horizontally what is the work done by gravity can we just say it is the gravitation 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 is not causing this displacement at all and therefore we can say the work done by gravity is zero because gravity is not the one that displays this body so make sense in similar manner imagine somebody is carrying a luggage on the 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 are pushing on the luggage up and which is perpendicular to the direction of the displacement and so we can say this force has no effect in 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 is 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 done in 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 distress one doesn't come into the picture during the vertical motion in gravity is doing work gravity is pulling and making displace so this matters again during the horizontal motion that displacement doesn't matter again then in the vertical motion it matters so during the horizontal motion since the distance the work done is zero we can just get rid of the displacement that distance one 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 that's what we meant earlier that 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 therefore the work that in this case is exactly the same has 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 on on a slide right how do we 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 again imagine the ball goes forward and comes down and so on and so forth what we the work done but again we don't have to worry about the horizontal displacements because the world in my gravity would be zero right in the perpendicular direction gravity has no effect so the only displacement that mattered 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 hatch so this means even if I make the staircase smaller the work done by gravity should still be the same so now let me make the staircase even smaller no work done should still be the same let's make the staircase even smaller the work done will still be the same now you can imagine if we make on make if we go on making a staircase smaller and smaller and smaller and smaller the motion of the ball will start becoming smoother and smoother and eventually if you imagine the staircase small is 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 it's going in a very tiny staircase from here to here and when you add up the all the displacements only the vertical displacements will matter and again the total displacement will still be the just hatch 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 we can say even if our ball was going in some crazy weird path and the work done by gravity will again be just the same so it seems like the work done by gravity doesn't depend upon what part it takes to go from one point to another it only depends upon the height that it covers all right the height is all that matters it does not depend on the path and this is some 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 they'll 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 0 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 the way one way to argue this is instead of imagining that it fell through a curve like this we can imagine it went through very very very tiny staircase and then when the ball goes forward the work done would be zero so those those displacements don't matter the only displacement that will come into our equation is the vertical displacement and when you add all of them up we saw it will 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