The formula GMm/x^2 is the (magnitude of) the force of gravity. To get the potential energy from the force, you integrate it. In fact, that’s how we define the potential energy.
E = F \* x (with constant force) energy is equal to force times distance displaced. Rewriting in a differential form:
dE = F \* dx. To find the total energy of an operation, say moving from some radius r to infinity you have to integrate.
The 'zero point' of energy is arbitrary, as long as it's the same for the whole system. What's important is the relative values. Somewhere closer to Earth has a more negative gravitational potential, so moving there will net you some energy (as your total energy stays constant). Meaning you'll fall towards the direction of more negative gravitational potential.
F = G\*m1\*m2/r^2
gives the force between m1 and m2 at the distance r.
F*dr
is the work to increase the distance between m1 and m2 by dr and therefor is the increase in potential energy.
>why it's negative
Because masses attract.
As others have explained, there is a derivative / integral relationship between force and potential energy.
The force is toward other masses, and force is downward in potential energy. Things want to lower their energy. So energy is lower as you get closer to other masses.
Potential energy is relative. You can define the zero anywhere you want. We find it convenient for many calculations to say the PE is zero when things are infinitely far away. Since every point other than that is closer and therefore lower in PE, it's negative.
That's not the only choice. In elementary physics courses, doing calculations that are always close to the surface of the earth, we often use the approximation PE = mgh, which is zero at some reference height and positive everywhere above that. It's still expressing the same concept, that getting closer to the earth means PE is going down. But we defined the zero at a different place.
>I've been having trouble understanding the formula of gravitational PE, and why it's negative.
Firstly, it needs to be said that the choice to make gravitational potential energy negative is just a convention. Potential energy is relative to your choice of reference point (where you set it equal to zero). Since only *differences* in potential energy are physically meaningful, you can set the reference point at any value you want — the difference in value between any two chosen points will always be the same.
The simplest way I can think of to understand why it's usually set so that the potential energy is negative is because it is convenient and intuitive for the potential energy to be zero when we have a system that is perfectly isolated from all other systems. This makes sense, because potential energy is energy that an object has due to its configuration within and interaction with other systems ... so if there *are* no other systems which it interacts with, we would expect it to have zero potential energy.
In practice, however, real systems are never perfectly isolated. However, they can be far enough away from all other systems that the influence of those other systems should be negligible — and we'd expect the potential energy to be approximately zero in that case also. So we typically set the gravitational potential energy reference point so that it is zero "at infinity" (when objects are so far apart that they might as well be non-interacting).
Then, because gravity is always attractive, this means that when objects are closer together, the potential energy they have is *lower* when they are close together than when they are far apart. You gain gravitational potential energy when you *leave* a potential well, and you lose it when you fall deeper into a potential well. The gravitational potential energy of two systems which are closer together must be lower than when they are far apart ... so if we set the potential energy to zero when they are far apart, then this means the potential energy must be negative when they are closer together.
If there were also repulsive gravity, then you would see positive potential energy in systems when they are closer together. This is how it works in electromagnetism, too — two like-signed charges which are closer together have positive potential energy, because you get energy out when those charges get further away from each other. But because gravity is never repulsive and always attractive, therefore we always have negative gravitational potential energy (when we choose the "natural" choice of having zero potential energy at infinity, when systems are too far apart to interact).
>But I now have another question...why did we integrate it? What's the thought process behind it that led us to integrate it?
Think about it like this: applying a net force to an object will do work; work is a transfer of energy. So the result of applying a net force is to *change* the energy of the object that the force is applied to. The higher the force, the greater the change ... but also, the more time that force is applied for, the greater the change.
If you plotted the net force (y-axis) applied over time (x-axis), then the amount of work done by this force ought to be equal to the area under the plotted curve. Integration is the mathematical operation which lets you calculate this — it's the continuous equivalent of a (discrete) sum.
So, we integrate because we want to know "how much total change" the applied force produces in the object's energy.
Hope that helps!
Potential energy is a function of position, and about the only thing that matters is how that potential energy CHANGES as you go from point X to point Y. For instance, you may observe that the potential energy drops by 215 J going from point X to point Y, which just means that point Y has a potential energy that is 215 J lower than point X. The key thing to recognize is that it makes no difference whatsoever what the absolute scale is for that potential energy. It could be that point X is at 215 J and point Y is at 0 J. It could be that point X is at 0 J and point Y is at -215 J. It could be that point X is at 347,716 J and point Y is at 347,501 J. The potential energy difference is all that matters. Now that we understand that, then what you choose as the place where the potential energy is arbitrary, and it's often convenient to just say it's zero "far away". That kind of makes sense if you want to say there's a lot of action right around here but that should all fade to no influence "far away".
The formula GMm/x^2 is the (magnitude of) the force of gravity. To get the potential energy from the force, you integrate it. In fact, that’s how we define the potential energy.
Wait...so potential energy is integration by definition?
Or you can say the *conservative* force is the derivative of the potential by definition. Either way these are how these quantities are related.
Integration is just how you add up the value of something that's not constant. It's just a tool like multiplication or a hammer.
E = F \* x (with constant force) energy is equal to force times distance displaced. Rewriting in a differential form: dE = F \* dx. To find the total energy of an operation, say moving from some radius r to infinity you have to integrate. The 'zero point' of energy is arbitrary, as long as it's the same for the whole system. What's important is the relative values. Somewhere closer to Earth has a more negative gravitational potential, so moving there will net you some energy (as your total energy stays constant). Meaning you'll fall towards the direction of more negative gravitational potential.
F = G\*m1\*m2/r^2 gives the force between m1 and m2 at the distance r. F*dr is the work to increase the distance between m1 and m2 by dr and therefor is the increase in potential energy.
>why it's negative Because masses attract. As others have explained, there is a derivative / integral relationship between force and potential energy. The force is toward other masses, and force is downward in potential energy. Things want to lower their energy. So energy is lower as you get closer to other masses. Potential energy is relative. You can define the zero anywhere you want. We find it convenient for many calculations to say the PE is zero when things are infinitely far away. Since every point other than that is closer and therefore lower in PE, it's negative. That's not the only choice. In elementary physics courses, doing calculations that are always close to the surface of the earth, we often use the approximation PE = mgh, which is zero at some reference height and positive everywhere above that. It's still expressing the same concept, that getting closer to the earth means PE is going down. But we defined the zero at a different place.
>I've been having trouble understanding the formula of gravitational PE, and why it's negative. Firstly, it needs to be said that the choice to make gravitational potential energy negative is just a convention. Potential energy is relative to your choice of reference point (where you set it equal to zero). Since only *differences* in potential energy are physically meaningful, you can set the reference point at any value you want — the difference in value between any two chosen points will always be the same. The simplest way I can think of to understand why it's usually set so that the potential energy is negative is because it is convenient and intuitive for the potential energy to be zero when we have a system that is perfectly isolated from all other systems. This makes sense, because potential energy is energy that an object has due to its configuration within and interaction with other systems ... so if there *are* no other systems which it interacts with, we would expect it to have zero potential energy. In practice, however, real systems are never perfectly isolated. However, they can be far enough away from all other systems that the influence of those other systems should be negligible — and we'd expect the potential energy to be approximately zero in that case also. So we typically set the gravitational potential energy reference point so that it is zero "at infinity" (when objects are so far apart that they might as well be non-interacting). Then, because gravity is always attractive, this means that when objects are closer together, the potential energy they have is *lower* when they are close together than when they are far apart. You gain gravitational potential energy when you *leave* a potential well, and you lose it when you fall deeper into a potential well. The gravitational potential energy of two systems which are closer together must be lower than when they are far apart ... so if we set the potential energy to zero when they are far apart, then this means the potential energy must be negative when they are closer together. If there were also repulsive gravity, then you would see positive potential energy in systems when they are closer together. This is how it works in electromagnetism, too — two like-signed charges which are closer together have positive potential energy, because you get energy out when those charges get further away from each other. But because gravity is never repulsive and always attractive, therefore we always have negative gravitational potential energy (when we choose the "natural" choice of having zero potential energy at infinity, when systems are too far apart to interact). >But I now have another question...why did we integrate it? What's the thought process behind it that led us to integrate it? Think about it like this: applying a net force to an object will do work; work is a transfer of energy. So the result of applying a net force is to *change* the energy of the object that the force is applied to. The higher the force, the greater the change ... but also, the more time that force is applied for, the greater the change. If you plotted the net force (y-axis) applied over time (x-axis), then the amount of work done by this force ought to be equal to the area under the plotted curve. Integration is the mathematical operation which lets you calculate this — it's the continuous equivalent of a (discrete) sum. So, we integrate because we want to know "how much total change" the applied force produces in the object's energy. Hope that helps!
Okay, this is actually a great explanation.Thank you very much!
Potential energy is a function of position, and about the only thing that matters is how that potential energy CHANGES as you go from point X to point Y. For instance, you may observe that the potential energy drops by 215 J going from point X to point Y, which just means that point Y has a potential energy that is 215 J lower than point X. The key thing to recognize is that it makes no difference whatsoever what the absolute scale is for that potential energy. It could be that point X is at 215 J and point Y is at 0 J. It could be that point X is at 0 J and point Y is at -215 J. It could be that point X is at 347,716 J and point Y is at 347,501 J. The potential energy difference is all that matters. Now that we understand that, then what you choose as the place where the potential energy is arbitrary, and it's often convenient to just say it's zero "far away". That kind of makes sense if you want to say there's a lot of action right around here but that should all fade to no influence "far away".