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# Gravitational potential energy derivation

## Video transcript

if I hold this bowling ball at some height then we say that the ball has gravitational potential energy and the reason we say that is because if I let go of that bowling ball then gravity can do work on that ball now the question we want to try and answer in this video is exactly how much potential energy does this ball have meaning if I know the mass of this ball let's say it's m and I know exactly how high it is what is the potential energy that's what I'm gonna try and figure out all right so how do we do this I think we can do that if we understand exactly the meaning of this word so what is the meaning of the word energy energy is the capacity to do work it's a number which tells how much work you can do so what would be gravitational potential energy it is the capacity of gravity to do work how much work gravity can do so over here if gravity can do let's say hundred joules of work in moving that ball down then we will say gravitational potential energy is hundred joules if gravity can do only two joules of work then we will say it's potential energy is only 2 joules okay so from this we can immediately say the gravitational potential energy by definition should equal how much work gravity can do on the ball gravity can do on the ball now before you begin you may be wondering why is it called potential what is this meaning of potential well in short what it means is if I were to let go of this ball then gravity will start pulling on it and that ball gains kinetic energy right it starts moving so it's basically saying that the ball has the potential to gain kinetic energy that's basic idea behind this and of course we have talked a lot about potential energies and kinetic energies in a previous video called energy all right so if you need more clarity on that great idea to go back and watch that video ok so let's calculate the on the ball how do I calculate work well in physics work done is force acting on an object multiplied by the displacement of the object so that means all we have to do is calculate how much force is acting on this ball multiplied by how much displacement this ball gets all right so how much force is acting on this ball which force are we talking about well since you are dealing with gravity we are talking about the gravitational force and we have seen before the force of gravity or any object is just mass times the gravitational acceleration mg we also call this as the weight of that ball okay and how much displacement can this ball get well if I just drop this ball the ball can will fall all the way from here to here and so the displacement of that ball is just going to be the height and so what do you think will happen if I multiply force and displacement what will I get can you just pause the video and think about this all right we'll get force which is the weight mg times displacement which is the height and so this is the gravitational potential energy of the ball it's M times G times H let's take some numbers let's say the mass of our bowling ball is 2 kilograms and imagine we raise it to a height of say 3 meters can you calculate how much potential energy this ball will have go ahead substituent see how much you'll get okay so if we substitute we'll get mass which is 2 kilograms I'm not going to write the unit's and the reason for that is because I already know the units of my potential energy since the energy is the same as work done the unit of energy will be the same as work done which is joules and everything is in standard units so our answer is going to be in joules ok so I'll not put in the unit's so 2 kilograms times G how much is g g is 9.81 we can take it as 10 we can out it off times the height which is 3 3 meters and how much is that give us that uses three toes are six six times ten is 60 so the potential energy of this ball is 60 joules and what does that mean this means that if I let go of this ball then gravity will do 60 joules of work in pushing that ball or pulling that ball from here to here now before we wrap up this video one doubt I always had is should I calculate this height always from the ground itself here's what I mean imagine I'm holding a bowling ball on the first floor of a building now to calculate the potential energy over here should I calculate the height from the floor of that room or should I calculate the height from the ground floor of that building which height should I take what do you think this was my biggest confusion well you can take any height you want it all depends upon the situation for example if you're dropping this ball on the first floor then it will go and hit the floor over here this is the lowest point right so this is kind of the ground for us in that case it just makes sense to say that this is the height and so we'll get a smaller value of potential energy and that means gravity can only do work from here to here so less potential energy on the other hand if you were to take that ball and throw it outside the balcony first of all you should never throw a bowling ball outside the balcony but let's say you did now that ball can fall all the way down to the ground so now it makes sense to say this is the height of that ball and so you'll get a larger potential energy for that same position because now gravity can do more work from here all the way till here so yeah your potential energy completely depends upon what you choose as your ground and you're free to choose whatever whichever level you want as your ground and what I like to do is whenever I'm calculating potential energy of any object I will take the lowest point that object can go as my ground and calculate the height from there so if I'm dropping the ball on the first floor itself then to calculate the potential energy of the ball over here I would just take this as the height and just to give you another example let's say we're dealing with a pendulum imagine that Bob is attached to a string and we can make it swing and now if I want to know what is the potential energy of the ball over here again I will think what is the lowest point that ball can come in this experiment so here if I let go of the ball notice the ball can swing to this this is the lowest point the ball can come after that the ball will go again go up there right so in this case it'll just make sense to say this is our ground level and this as the height to calculate the potential energy of the ball at this point so to summarize what did we learn in this video we learned how to calculate the potential energy of anybody due to gravity and how do we do that well we calculate how much work gravity can do and just like any work we multiply force into displacement the force over here is the force of gravity and the displacement will just be the height giving us MGH which means if you ever forget this equation no problem if you just go back to the basics of work done you can always derive it in one step and if you ever get confused about from where should you calculate that height where is your ground in general and you're free to choose whichever level as you want as your ground and what I like to do is that in any experiment I ask myself what is the lowest point that object can reach in that experiment and then call that as my ground and calculate the height from there to calculate ench of energy