# Variation of potential energy with height

• jasonchiang97
In summary, the given potential function V = mgz(1+z/re) accounts for the variation of gravity with height by considering the potential energy of an object relative to what it would have at the Earth's surface. This can be seen by subtracting the potential energy at the surface of the Earth, -mMG/R, from the formula. Additionally, the force given by this potential function is F = -mgz(1-z/R).

## Homework Statement

Show that the variation of gravity with height can be accounted for approximately by the following potential function

V = mgz(1+z/re)

in which re is the radius of the Earth. find the force given by the above potential function.

V = GM/r

## The Attempt at a Solution

Let z be a point above the surface of the Earth and R be the radius of the earth

V(z) = -GM/(R+z) = (-GM/R)(1+z/R)-1

Using the binomal expansion of (1+x)n ≈ 1 +nx +n(n-1)x2/2

I get

V = (-Gm/R)(1-z/R + (z/r)2

Expanding gives

V = -GM/R + GMz/R2 - GMz2/R3

but the force is given by

F = GM/R2 = mg

replace GM/R2 with mg to get

V = -mgR + mgz - mgz2/R

= -mgR + mgz(1-z/R)

I'm not sure why I have an extra -mgR term.

Gravitational potential energy, like gravitational and electrical potential, has no absolute level, so we can add any constant we like to it.
Gravitational potential energy near the Earth's surface is usually given a gauge (choice of a constant added) such that te value at the Earth's surface is 0. In that gauge, the gravitational PE of an item of mass m that is z metres above the Earth's surface, which is distance R from the centre, is the formula you got, minus the grav PE it would have at the Earth's surface, which is -mMG/R. Taking the difference removes the unwanted term.

Strictly speaking, the formula is for the grav PE of the mass relative to what it would have at the Earth's surface.

By the way, the function given in the OP is of Potential Energy, not potential. To get potential we have to divide by m.

Okay, so I multiply by m to get the potential energy, but I don't understand what you mean by adding a constant. So my first term in my expansion should just be 0 somehow?

jasonchiang97 said:
Okay, so I multiply by m to get the potential energy, but I don't understand what you mean by adding a constant. So my first term in my expansion should just be 0 somehow?
Your expansion is fine. But the potential energy your formula gives is relative to what the object's PE would be at the centre of the Earth. The formula you are asked to prove is for PE relative to what it would be at the surface of the Earth. Since the latter is -mMG/R, you subtract that from what you got, to get the PE relative to the Earth's surface.

I see. Thanks!

## 1. What is potential energy?

Potential energy is the energy that an object possesses due to its position or configuration. It is stored energy that has the potential to do work.

## 2. How does potential energy vary with height?

Potential energy varies with height according to the equation PE = mgh, where m is the mass of the object, g is the acceleration due to gravity, and h is the height of the object. As the object's height increases, its potential energy also increases.

## 3. Why does potential energy increase with height?

This is because the force of gravity is acting on the object, and as the object is lifted to a higher height, the force of gravity does work on it, increasing its potential energy. The higher the object is lifted, the more potential energy it gains.

## 4. What is the relationship between potential energy and kinetic energy?

Potential energy and kinetic energy are forms of energy that are interrelated. Potential energy can be converted into kinetic energy, and vice versa. For example, when an object is dropped from a height, its potential energy is converted into kinetic energy as it falls.

## 5. Does all potential energy have to be converted into kinetic energy?

No, not all potential energy has to be converted into kinetic energy. An object can have potential energy without any kinetic energy, such as a book sitting on a shelf. Additionally, potential energy can be stored in other forms, such as chemical potential energy in a battery or elastic potential energy in a stretched spring.