Calculating Potential Energy

As we know, an object can store energy due to its position. In the case of a bow and an arrow, when the bow is drawn, it stores some amount of energy, which is responsible for the kinetic energy it gains when released.

Similarly, in the case of a spring, when it is displaced from its equilibrium position, it gains some amount of energy which we observe in the form of stress we feel in our hands upon stretching it. We can define potential energy as a form of energy that results from the alteration of its position or state.

After understanding potential energy and potential energy definition, let us learn the formula, unit, and examples of potential energy. In this article, let us learn potential energy examples in detail.

The potential energy formula

Let's look under the hood of the potential energy calculator. To help you picture it, our example will be the massive wrecking ball on a crane. The gravitational potential energy of this ball depends on two factors - the mass of the ball and the height it's raised to. The relationship between gravitational potential energy and the mass and height of an object is described by the following equation:

PE grav.=m × h × g

Where:

PE grav. - Gravitational potential energy of an object;
m - Mass of the object in question;
h - Height of the object; and
g - Gravitational field strength acting upon the object (1 g or 9.81 m/s2 on Earth).

The formula is relatively simple. An object which is not raised above the ground will have a height of zero and, therefore, zero potential energy. When you double the mass or the height of an object, its potential energy will also double.

Example

Calculating Potential Energy

Potential energy is the stored energy of an object due to its position or state. The goal of calculating potential energy is to determine how much energy an object has based on its height and mass.

The general approach to calculating potential energy includes:

Identifying the mass of the object.
Knowing the height of the object above a reference point.
Applying the formula for potential energy to calculate the result.

Potential Energy Formula

The general formula for gravitational potential energy is:

\[ PE = mgh \]

Where:

PE is the potential energy (in joules, J).
m is the mass of the object (in kilograms, kg).
g is the acceleration due to gravity (9.8 m/s² on Earth).
h is the height of the object above the reference point (in meters, m).

Example:

If a 5 kg object is placed 10 meters above the ground, the potential energy is:

Step 1: Multiply the mass by gravity: \( 5 \times 9.8 = 49 \).
Step 2: Multiply by the height: \( 49 \times 10 = 490 \, J \).

Types of Potential Energy

Potential energy can exist in different forms, including:

Gravitational Potential Energy: Energy stored in an object due to its height.
Elastic Potential Energy: Energy stored in stretched or compressed elastic materials, such as springs or rubber bands.
Chemical Potential Energy: Energy stored in chemical bonds, such as in food or fuel.

Real-life Applications of Potential Energy

Calculating potential energy has many practical applications, such as:

Understanding how much energy is stored in objects at different heights (e.g., a roller coaster at the top of a hill).
Designing energy-efficient systems (e.g., hydroelectric power plants using water stored at high elevations).
Analyzing energy in sports (e.g., determining the energy in a high jumper before they take off).

Common Units of Potential Energy

SI Unit: The standard unit of potential energy is the joule (J).

Other units may include calorie (used in food energy) and electron volts (used in atomic physics), but the joule is the most common unit in physics calculations.

Calculating Potential Energy Examples Table
Problem Type	Description	Steps to Solve	Example
Calculating Gravitational Potential Energy	Finding potential energy due to an object's height above the ground.	Identify the mass \( m \), gravitational acceleration \( g = 9.8 \, \text{m/s}^2 \), and height \( h \). Use the potential energy formula: \( PE = mgh \).	For an object with mass \( 10 \, \text{kg} \) at a height of \( 5 \, \text{m} \), the potential energy is \( PE = 10 \times 9.8 \times 5 = 490 \, \text{J} \).
Potential Energy in Springs	Finding the elastic potential energy stored in a compressed or stretched spring.	Identify the spring constant \( k \) and the displacement \( x \) from the equilibrium position. Use the elastic potential energy formula: \( PE = \frac{1}{2} k x^2 \).	If a spring with \( k = 200 \, \text{N/m} \) is stretched by \( 0.3 \, \text{m} \), then \( PE = \frac{1}{2} \times 200 \times (0.3)^2 = 9 \, \text{J} \).
Comparing Potential Energy at Different Heights	Understanding how potential energy changes as height changes.	Calculate the potential energy at different heights using \( PE = mgh \). Compare the values to understand how energy changes with height.	A 2 kg book on a shelf at 3 m has \( PE = 2 \times 9.8 \times 3 = 58.8 \, \text{J} \), but at 1 m, it has \( PE = 19.6 \, \text{J} \).
Real-life Applications	Applying potential energy concepts to solve practical problems.	Calculate the energy stored in a raised object. Determine the energy required to lift an object.	If a person lifts a 15 kg box to a height of 2 m, the stored energy is \( PE = 15 \times 9.8 \times 2 = 294 \, \text{J} \).