# Gravitational Potential Energy

### Table of Contents

- What is Potential Energy?
- Conservative Force
- How do you find Gravitational Potential Energy (GPE)?
- Gravitational Potential Energy
- Change in Gravitational Potential Energy
- Gravitational Potential Energy of a two particle system
- Gravitational Potential Energy of a three-particle system

## What is Potential Energy?

The energy stored in a body by virtue of its position or configuration is called **Potential Energy**.

In the 19th century, William Rankine invented the term Potential Energy.

Consider, a body of mass m placed on the surface of the earth. The weight of the body surface on the earth is denoted by weight = mg. Then raise the body through height h above the surface the minimum force required is its weight.

Work done to raise the body above the surface is stored in the body as it’s potential energy.

The gravitational potential energy of the body as a function of height h is denoted by v(h) and it is the negative of work done by the gravitational force while raising the body through the height.

v(h) = mgh

If h is the variable, it can be easily shown the gravitational force F is equal to the negative derivative of v(h) with respect to h. Thus

The negative sign indicates that the gravitational force is directed downwards(Fig:2). Hence, when release the body comes down with increasing speed converting its potential energy into kinetic energy.

The body strikes the ground with the velocity v denoted by

v^{2} = 2gh

On multiplying both sides the above equation with half m, we get

Thus, this proves the potential energy of the body transform to its kinetic energy when the body is released. Hence physically the concept of potential energy is applicable type of forces where worked done against the force is stored in it.

**Mathematically**, the potential energy v(x) is defined for any force F(x), which can be written as:

On further simplification, we can arrive at the conclusion

The work done on a body by the force depends on the initial and final position only and not on the path followed by the body. Such a force is called **Conservative Force**.

## Conservative Force

If the work done by a force is independent of the path followed then it is a conservative force.

### Example of Conservative Force

Gravitational force and elastic force in the case of stress spring etc.

When a box is dropped from a certain height, the work done by the gravity on the box is w = mgh.

When the same box slide down smooth in an inclined plane at a inclination of θ with horizontal, the force acting down along the incline is mg sinθ. The distance traveled by the box is I and hence the work done by gravity on it is denoted by the expression

W = mg sin θ (AC) = mg sin θ l = mgl sin θ

l sin θ = h(According to Geometry)

W = mgh

## How do you find Gravitational Potential Energy (GPE)?

Gravitational Potential Energy can be defined as the energy of a body which is arising out of the force of gravity that means when the gravitation force is acting on a body and that force is resulting a kind of potential energy than the potential energy known as gravitational potential energy. Now, if the position changes on account of force then change in potential energy is the amount of work done on the body by the force. It means suppose we have any objects A and another B. Now if the position of B changes only because of a particular force. Let’s assume M is changing its position due to the gravitational force then the change in potential energy. Because of the change in the position of the object is equal to the amount of work done by the gravitational force.

So, this change in potential energy passing through from P to Q will be equal to the work done by the gravitational force.

Assume P is position 1 and Q is position 2, in that case, the work done on taking the object from position 1 to position 2 is equal to the change in potential energy that is the potential energy at point 2 minus the potential energy at point 1.

## Gravitational Potential Energy

When a body is lifted to a height from the surface of the earth, the work done on the body is denoted by expression,

**W = mgh, where m is the mass of the body.**

When a body is in the ground label, the work done on the body is zero as its height is zero.

**W = 0 (h = 0)**

Thus, the ground is taken as the reference level or Datum level. The work is done on the body while lifting is stored in it in the form of gravitational potential energy.

If the position of a body is changed by the application of a force. The change in the potential energy of the body is equal to the amount of the work done on the body.

Now consider a body moved vertically from a point at a height h_{1 }to another point h_{2 }as shown. The difference in the gravitational potential energy at this point is equal to work done by the force applied on it while lifting it through the height between the two points.

**W = mgh _{2} – mgh_{1}**

## Change in Gravitational Potential Energy

The concept of change in gravitational potential energy of a body is calculated by the product of its mass, the acceleration due to gravity at a given place and the change of the height of the body from the chosen reference level. That means,

**Gravitational Potential Energy = m × g × (h _{2} – h_{1}) …. (1)**

This is not applicable when we consider a point very far from the surface of the earth as the value acceleration due to gravity is no more constant. Thus the expression m,g in the equation (1) as before acting on the body due to the gravitational pull of the earth is not applicable at that distance.

To understand the change in gravitational potential energy of a two-particle system. We use a gravitational force between the two and the concept of Integral Calculus.

Now consider a particle of mass m_{1} placed at a fix position A, and another of a mass m_{2} moved from position B to C as shown in the picture.

In moving the second particle from B to C, consider orbitary position D, where it’s distance from A at a given moment is r. The gravitational force of attraction between particle of masses m_{1 }and m_{2}. Separated by distance r which along DA is

If the particle is moved through an infinitesimally small distance dr. The work done by the gravitational force on the second particle is denoted by -Fdr.

dW = -Fdr

On substituting F in this expression, we get,

The negative sign in the equation is due to the fact that the direction of the displacement is opposite to that of the force. By definition, the negative work done on the second particle is equal to the change in gravitational potential energy of the two particle system during this small displacement.

**If the particle is moved through an infinitesimally small distance dr. The work done by the gravitational force on the second particle is denoted by – Fdr**

Thus,

The change in gravitational potential energy of the two particle system as the second particle moves from B to C is a function of distance r, is denoted by

Now consider the distance between two particles is infinity, in this case the potential energy of the system is chosen to be zero.

**U (∞) = 0**

**In general, U(r) = U(r _{2}) – U(r_{1}) = U(r) – U(∞)**

## Gravitational Potential Energy of a two particle system

In general, why defining the gravitational potential energy of a two particle system. We bring a particle from infinity close to another particle. In this expression r_{1} is infinity and r_{2 }is equal to r. By moving the second particle from infinity to close to the first particle.

Here

**U(r) = U(r _{2}) – U(r_{1}) = U(r) – U(∞)**

As U (∞) = 0. Then the same expression can be written as:

**U (r) = U (r) - 0**

Thus, the gravitational potential energy of the two particle system separated by distance r is denoted by

In this system, if the mass of one particle, say m_{1 }= Unity and

m_{2} = m

Then the energy of two particles system is

This equation is defined as the gravitational potential at a distance r from particle of mass m.

So far, we have considered a two particle system.

## Gravitational Potential Energy of a three particle system

Now, if we considered a three-particle system, say at position P, Q and R as shown we have three pairs of particles PQ, PR and QR.

** **Thus, the total gravitational potential energy of the three particle system is the sum of the potential for the three pairs of particles,

Applying superposition principle to an isolated system containing n number of particles. The total potential energy of the system particles is equal to the sum of the energies of all the possible pairs of its constituent particles.

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