## Calculating potential energy

We will learn what potential energy is in this article as well as how to calculate various types of potential energies, including gravitational, electrical, and others.

It is important to note here that when we talk about the term potential energy it does not automatically means that it is mechanical in nature. As we would see later on in this article that the type of potential energy depends on the nature of the force being applied.

The most important thing to know about potential energy is that it is related to the position of the object under consideration.

### What is Potential energy?

Potential energy is the amount of energy that an object has because of its position in space.

It is determined by the object’s mass, position, and type of forces that are acting on it. For example, the potential energy of an object on a high shelf is higher than that of an object on the ground. This is because the object on the shelf is further away from the ground and has more potential to move.

Examples of potential energy in physics include gravitational potential energy, elastic potential energy, electrical potential energy, and chemical potential energy.

### Why do we need to calculate potential energy?

We need to calculate potential energy because it measures an object’s ability to do work.

Work is the amount of energy transferred from one object to another, and potential energy is the energy that can be used to do work.

Knowing the potential energy of an object can help us determine how much work can be done on it, or how much energy it can release when it moves.

## Calculating different types of Potential Energy

Different types of potential energy, such as gravitational, electrical, and spring-mass systems, have different formulas for calculating their potential energy.

In this section, we will explore how to calculate the potential energy of each of these different systems.

### Calculating Gravitational PE

Gravitational potential energy is the energy an object has due to its position in a gravitational field.

It is equal to the work done to move the object to its current position against the force of gravity.

The formula for gravitational potential energy is

$$U = mgh$$where,

$U$ is the letter used to represent potential energy.

$m$ is the mass of the object,

$g$ is the acceleration due to gravity, and

$h$ is the height of the object above a reference point.

Now that we know the formula for gravitational potential energy lets us use this formula to learn how to calculate gravitational PE with the help of an example.

#### Question:

Consider a hammer that weighs 2.2 kg and falls 50 m to the ground from a window ledge. How much potential energy does it have?

#### Solution:

**Step 1:** Read the question carefully and identify the information regarding various physical quantities provided in the question.

It is given in the question that

- the mass of the hammer $m=2.2 \, Kg$
- If we take the surface of Earth as zero potential reference point then the height of the hammer above the Earth’s surface is $h=50\, m$
- Acceleration due to gravity $g=9.8m/s^2$

**Step 2:** Plug all these values into the formula for gravitational potential energy.

$U=mgh=(2.2\,Kg)\times(50\, m)\times(9.8\,m/s^2)$

**Step 3:** Calculate the results in the units of energy

So,

$U=(2.2\,Kg)\times(50\, m)\times(9.8\,m/s^2)=1078\,Kg.m^2/s^2=1078\,J$

The gravitational potential energy of the hammer is $1078\, J$

### Calculating Elastic PE

Elastic potential energy is the energy stored in an object when it is stretched or compressed. It is released when the object returns to its original shape.

For example, when you stretch a rubber band, it stores energy that is released when you let go of the rubber band and it snaps back to its original shape.

The formula for elastic potential energy is

$$U = \frac{1}{2}k\Delta x^2$$

where,

- $U$ is the elastic potential energy in Joules,
- $k$ is the constant of proportionality and is known as the spring constant.
- $\Delta x$ is the displacement of the spring from its equilibrium position and is measured in
*meters*.

Now that we have the formula for elastic potential energy, let us use it to understand how to calculate elastic PE using an example.

#### Question:

What is the elastic potential energy stored in a spring of spring constant $k=400 N/m$ when a boy pulls it and it stretches by $0.20m$?

#### Solution:

**Step 1:** Read the question carefully and identify the information regarding various physical quantities provided in the question.

It is given in the question that

- deformation in the spring $\Delta x=0.20 \, m$
- Spring constant of the spring is $k=400 N/m$

**Step 2:** Plug in all these values into the formula for elastic potential energy.

$U = \frac{1}{2}k\Delta x^2=\frac{1}{2}\times (400N/m)\times(0.20\,m)^2$

**Step 3:** Calculate the results in the units of energy

So,

$U=\frac{1}{2}\times (400N/m)\times(0.20\,m)^2=8\,J$

The elastic potential energy stored in the spring is $8\,J$

### Calculating Electric potential energy

Electric potential energy is the energy possessed by a system of charges by virtue of their position.

The electric potential energy of a system of point charges may be defined as the amount of work done in assembling the charges at their locations by bringing them in, from infinity.

The electric potential energy of a system of charges is the sum of the potential energies of the individual charges.

If two charges $q1$ and $q2$ are separated by a distance $r$, the electric potential energy of the system is given by the formula.

$$U =\frac{1}{4\pi \epsilon_0}\cdot \frac{q_1q_2}{r^2}$$

where,

- $U$ is the electric potential energy,
- $\epsilon_0$ is the permittivity of vacuum

Now that we have the formula for electric potential energy, let us use it to understand how to calculate electric PE using an example.

#### Question:

Calculate the electrostatic potential energy of a system consisting of two charges $7\mu C$ and $-2\mu C$ (and with no external field) placed at $(-9\, cm, 0,0)$ and $(9\, cm,0,0)$ respectively.

#### Solution:

**Step 1:** Read the question carefully and identify the information regarding various physical quantities provided in the question.

It is given in the question that

- charge $q_1=7\mu\,C=7\times 10^{-6}C$
- charge $q_2=-2\mu\,C=-2\times 10^{-6}C$
- distance between these two charges is $r=18\,cm=0.18m$
- $\frac{1}{4\pi \epsilon_0}=9\times 10^9\,Nm^2C^{-2}$

**Step 2:** Plug in all these values into the formula for electrostatic potential energy.

$U =\frac{1}{4\pi \epsilon_0}\cdot \frac{q_1q_2}{r^2}=\frac{9\times 10^9\times7\times10^{-6}\times(-2)\times10^{-6}}{0.18}$

**Step 3:** Calculate the results in the units of energy

So,

$U==\frac{9\times 10^9\times7\times10^{-6}\times(-2)\times10^{-6}}{0.18}=-0.7\,J$

The electrostatic potential energy stored in the spring is $-0.7\,J$

A negative potential energy indicates that work must be done against the electric field to separate the charges.