# How does potential energy increase, if height increases?

• sawer
In summary, the conversation discusses the relationship between potential energy and height in a gravitational field. It is noted that the potential energy is negative due to the gravitational potential energy formula, which ensures that the potential energy is lowered as two bodies come together. However, in the approximation V = mgh, the potential energy is proportional to h and the zero of energy is redefined as being at the surface of the Earth. This leads to a contradiction, which can be resolved by using an approximation for 1/(1+x) and showing that both expressions can be true.

#### sawer

(constants're ommitted)

1-) M=50, m=5, h=5 then Potential Energy = (50*5)/5 = 50

After increasing height

2-) M=50, m=5, h=10 then Potential Energy = (50*5)/10 = 25

Field strength decreases amount of h^2 so according to formulas potential energy decreases as h increases.

But that mustn't be true. But as I showed according to Potential energy and Field strength formulas, potential energy decreases as h increases. What is wrong here?

In that format, the gravitational PE is negative:

##PE = -\frac{GMm}{r}##

Alternatively, for a constant gravitational field, you can have:

##PE = +mgh##

sawer said:
(constants're ommitted)

1-) M=50, m=5, h=5 then Potential Energy = (50*5)/5 = 50

After increasing height

2-) M=50, m=5, h=10 then Potential Energy = (50*5)/10 = 25

Field strength decreases amount of h^2 so according to formulas potential energy decreases as h increases.

But that mustn't be true. But as I showed according to Potential energy and Field strength formulas, potential energy decreases as h increases. What is wrong here?

If you write things in the form that you have: V = -G m_1*m_2/r -- there is a very important negative sign that you lost, which you need to keep. What this does is ensure that the potential energy of two bodies at infinite separation is equal to zero, and that as the bodies come together, the potential energy is lowered (more negative). You could tell that you had made a mistake, because as you increased the height, the potential energy went down.

In the other common way of looking aqt things, close to the Earth's surface, you are using the simple approximation that V = m * g * h. In using this, you have done a number of things. One, you have redefined the zero of energy -- completely ok to do -- as being at the surface of the Earth ==> V = 0 when h = 0. You have also changed the relationship between distance. In the first case, the answer is proportiional to 1/r, in the second case, it is proportional to h.

Can you show how both of these expressions could possibly be true? Hint: Use the first expression, and let r = Re + h, where Re is the radius of the earth. Use an approximation for 1/(1+x), that is valid for small x (x<<1)

## 1. How does potential energy increase with an increase in height?

As an object is lifted to a higher position, it gains potential energy. This is because the object now has a greater potential to do work due to its increased height above the ground. The higher the object is lifted, the more potential energy it has.

## 2. What is the relationship between height and potential energy?

The relationship between height and potential energy is direct. This means that as height increases, potential energy also increases. Similarly, as height decreases, potential energy decreases.

## 3. How does potential energy change if an object is raised to a higher height?

If an object is raised to a higher height, its potential energy increases. This is because the object now has a greater potential to do work due to its increased distance from the ground.

## 4. What factors affect the potential energy of an object at a certain height?

The potential energy of an object at a certain height is affected by its mass and the strength of the gravitational force. A heavier object will have more potential energy at the same height compared to a lighter object. Additionally, the strength of the gravitational force at a particular location can also affect the potential energy of an object at a certain height.

## 5. Does potential energy increase indefinitely with an increase in height?

No, potential energy does not increase indefinitely with an increase in height. Once an object reaches a certain height, it will have a maximum potential energy. This is because the potential energy of an object is directly proportional to its height, and there is a limit to how high an object can be lifted.