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# Potential Energy and Infinity

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## Potential Energy and Infinity

We have come across **potential energy** in gravitational fields ever since GCSE. The higher you lift an object the more potential energy it has because the more work you had to do to put it there. Simple!! We even know an equation for it:

**E _{p} = mgh**

Now we have to consider a couple of problems.

First, in radial fields, g varies. So we can now longer use the above equation.

Secondly, we need to think about **where E _{p} is zero.** To answer this, read on!

**What causes objects to have potential energy?**

Objects in a gravitational field have the potential to do work if they have a force acting on them.

**Do objects on the Earth's surface have zero potential energy?**

No - they definitely have a force on them because they don't drift away so they have the potential to do work. In fact, push them into a hole and you will see some of that energy being converted into kinetic energy.

**Where is the gravitational force between objects zero?**

The force between objects due to gravity reduces as the distance separating the objects increases. So the force finally becomes zero at infinity.

Going back to our GCSE ideas, at the greatest possible distance from Earth, you should have your greatest possible value of E_{p}. But at infinity, the attraction due to g-field is **zero**. No force → no potential energy.

So the greatest E_{p} value you can get is **zero!!!!** And you get it at infinity.

When you move back towards Earth from infinity, your E_{p} reduces. So it must become less than zero! It must become **negative.**

Now that's confusing! The good news is, you don't have to know exactly where infinity is to answer any questions. That's because:

- We are often only interested in the change in potential energy rather than the actual value.
- Someone very thoughtfully came up with a way of calculating the potential energy that an object has by looking at the separation between the object and the centre of the field, rather than infinity. (Read on for the equation.)