Main content

# Gravitational potential energy derivation

Let's derive expression for gravitational potential energy. Created by Mahesh Shenoy.

## Want to join the conversation?

- I don't understand please annunciate.(2 votes)
- in the example of the pendulum, what force is doing the work??

the least i can make out is that it isnt gravitational force that is doing the work of making the pendulum move because the acceleration or displacement of the pendulum is sideways

and not downward, am i correct and someone please answer my question(1 vote)- It is the gravitational force only which does the work.

I appreciate your assumption that it must not be gravitational force as the motion here is sidewards. However, the downward pull exerted by the gravitation here acts sidewards as the pendulum is connected to a rigid end and can't come downwards.

What I say maybe doesn't make absolute sense, because the rigorous explanation of your question requires the mathematical understanding of vectors and vector components, which you will learn in the Class 11 physics course

Hope that helped. Feel free to comment if you need further help :)(1 vote)

- I think that she means the higher an object is,the more gravitational potential energy it has.(1 vote)
- wont the actual P.E. be 55.8?

60 is way off...(1 vote) - Gravity is acting downwards and displacement of the ball is upwards.There are in opposite direction.Then P.E = -work right? Please clear my doubt.(1 vote)
- Hey! If the value of g is calculated with the distance taken from the center of the earth, shouldn't the height be considered from the center of the earth too?(1 vote)
- why would the displacement in the pendulum be taken as the highest point and lowest point, clearly it has had much more spatial displacement. so the net force is going to be in the direction of the displacement aka around the hypotenuse of that 'h' isnt it?(1 vote)

## Video transcript

- [Narrator] 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. And the question you 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 you want
to 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 100 joules of work in moving that ball down, then we will say the
gravitational potential energy is 100 joules. If gravity can do only two joules of work then we will say it's potential
energy is only two 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 we 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
would 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. Okay, so let's calculate
the work done on the ball. How do I calculate work? Well in physics work done
is force acting on an object multiplied by the
displacement of that 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 we are dealing with gravity, we are talking about
the gravitational force and we have seen before
the force of gravity on any object is just mass times the gravitational acceleration, M G. 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 we just pause the
video and think about this? All right, we'll get force,
which is the weight M G, 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 two kilograms and imagine we raise it to a
height of, say, three meters. Can you calculate how
much potential energy this ball will have? Go ahead, substitute in and
see how much you'll get. Okay, so if we substitute we'll get mass which is two kilograms and now I'm not gonna write the units 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. Okay, so I'm not putting the units. So two kilograms times G. How much is G? G is 9.8, but we can take it as 10. We can round it off. Times the height which is three, three meters. And how much does that give us? That gives us, three twos are six, six times 10 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, 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'll 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 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 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 the
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 will just take this as the height. And just to give you another example, let's say we're dealing with a pendulum. Imagine that ball 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 go
again, go up there, right? So in this case, it
just makes sense to say this as our ground level and this as the height to calculate the potential energy of
the ball at this point. So to summarize, what have
we learned in this video? We learnt how to calculate
the potential energy of any body 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 M G H. 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? Then you are 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 the object can reach in that experiment? And then call that as my ground and calculate the height from there to calculate potential energy.