# Derivation of Potential Energy

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## What is Potential Energy?

Potential energy is the energy possessed by an object due to its relative stationary position in space, stress, or electric charge. Potential energy is the inherent energy of the body relative to its static position to the other objects. Potential energy is one of the two main types of energy, while the other is kinetic energy. The two types of potential energy are elastic potential energy and gravitational potential energy.

Elastic Potential Energy: Elastic Potential Energy is the energy present in objects that can be stretched or extended, such as trampoline, rubber bands, and bungee cords. The further an object can expand, the more elastic potential energy it has. Many items are designed specially to store elastic potential energy such as a twisted rubber band that powers a toy plane or a Coil spring of a wind-up clock.

The elastic potential energy formula derivation is:

U = 1/2 kx2

Where,

U = elastic potential energy

k = spring force constant

x = string stretch length in m

Gravitational Potential Energy: Gravitational potential energy is the energy acquired by an object due to a shift in its position when it is present in a gravitational field. In simple terms, it can be stated that gravitational potential energy is an energy that is linked to gravity or gravitational force.

The gravitational potential energy equation is:

GPE = m × g × h,

m = mass in kilograms,

g = acceleration (9.8 ms-2 on Earth)

h = height.

### Derivation for Potential Energy

The derivation of potential energy is discussed here. Potential energy is determined as the energy that is held by an object because of its stationary position. Joule is the S.I. unit of potential energy; its symbol is J. Scottish engineer and physicist William Rankine coined the term potential in the 19th century. The potential energy formula depends on the force enacting on two objects. The formula of gravitational potential energy is:

W = m × g × h = mgh

m = mass in kilograms

g = acceleration due to gravity

h = height in meters.

### Gravitational Potential Energy Derivation Equation

Let us consider an object, of mass M, which is placed along the x-axis, and there is a test mass m at infinity. Work done at bringing it without acceleration through a minimal distance (dx) is given by:

dw = Fdx

Here, F is an attractive force and towards the negative x-axis direction is the displacement. Therefore, F and dx are in a similar direction.

$dw = (\frac{GMm}{x^{2}}) dx$

Integrating both sides,

$w = \int_{r}^{\infty} \frac{GMm}{x^{2}} dx$

$w = -[ \frac{GMm}{x}]$

$w = -[\frac{GMm}{r}] - (\frac{-GMm}{\infty})$

$w = \frac{-GMm}{r}$

As the potential energy is stored as U, the gravitational potential energy at ‘r’ distance from the object having mass ‘M’ is:

U = - GMm/r

Now if another mass inside the gravitational field moves from one point inside the field to another point of the field of mass M, the other mass experiences a change in potential energy given by:

ΔU = GMm (1/ri – 1/rf)       [ ri= initial position and rf= final position ]

If ri > rf  then ΔU is negative.

### Derive an Expression for Gravitational Potential Energy at Height ‘h’

Let‘s consider an object taken to a height ‘h’ from the surface of the earth.

ri = R and rf = R + h

then,

ΔU = GMm [1/R – 1/(R+h)]

ΔU = GMmh/R(R + h)

When, h<<R, then, R + h = R and g = GM/R2.

On substituting this in the above equation we get,

Gravitational Potential Energy ΔU = mgh.