Gravitational Potential Energy and Kinetic Energy

1. Gravitational Potential Energy (GPE)

What is Gravitational Potential Energy?

Gravitational potential energy is the energy that an object has because of its position in a gravitational field. When you lift something up, you give it the ability to do work because of its height. For example, a book on a high shelf has more gravitational potential energy than the same book on the floor.

• If you lift a mass up, it gains gravitational potential energy
• If a mass falls down, it loses gravitational potential energy

We measure GPE from a reference point. On Earth's surface, we usually say that ground level has zero GPE.

Deriving the Formula: ∆EP = mg∆h

Let's work out where the formula for gravitational potential energy comes from. We start with the equation for work done:

W = Fs

Where:

• W = work done (in joules, J)
• F = force applied (in newtons, N)
• s = distance moved in the direction of the force (in metres, m)

Now imagine you're lifting an object of mass m through a vertical height ∆h (the symbol ∆ means "change in").

To lift the object, you need to apply an upward force to overcome its weight. The weight of the object is:

Weight = mg

Where:

• m = mass (in kilograms, kg)
• g = gravitational field strength (approximately 9.81 N/kg on Earth's surface)

When you lift the mass through a height ∆h, the distance moved in the direction of the force is equal to ∆h.

Using W = Fs, the work done becomes:

W = mg × ∆h

Since you've lifted the mass, it now has extra energy stored because of its new position. This extra energy is equal to the work you did lifting it. We call this energy change the change in gravitational potential energy:

∆EP = mg∆h

This is the formula for the change in gravitational potential energy in a uniform gravitational field (a gravitational field where g stays the same, which is true near Earth's surface).

Using the Formula ∆EP = mg∆h

The formula tells us:

• ∆EP = change in gravitational potential energy (J)
• m = mass (kg)
• g = gravitational field strength (N/kg), approximately 9.81 N/kg on Earth
• ∆h = change in height (m)

Important points:

• This formula only works in a uniform gravitational field – that means g must be constant. This is true near Earth's surface, but not in space where g varies significantly.
• We're measuring the change in GPE (∆EP), not the total GPE. This is because we can choose any reference point to be "zero height."
• The potential energy at ground level is taken to be zero

Relationship between GPE and height:

GPE increases linearly with height. This means if you double the height, you double the GPE. If you plot a graph of GPE against height, you get a straight line starting from the origin.

Example Calculation

Question: A person with mass 74 kg climbs five flights of stairs. Each flight has a height of 3.7 m. What is the person's gain in gravitational potential energy? (Use g = 9.81 N/kg)

Solution:

Step 1: Find the total change in height

• ∆h = 5 × 3.7 = 18.5 m

Step 2: Use the formula ∆EP = mg∆h

• ∆EP = 74 × 9.81 × 18.5
• ∆EP = 13,430 J ≈ 13,000 J

The person gains approximately 13,000 J of gravitational potential energy.