# 3.6.5 Gravitational Potential Energy

### Gravitational Potential Energy (Eₚ)

• Gravitational potential energy (Ep) is energy stored in a mass due to its position in a gravitational field
• If a mass is lifted up, it will gain Ep (converted from other forms of energy)
• If a mass falls, it will lose Ep (and be converted to other forms of energy)
• The equation for gravitational potential energy for energy changes in a uniform gravitational field is:

Gravitational potential energy (GPE): The energy an object has when lifted up

• The potential energy on the Earth’s surface at ground level is taken to be equal to 0
• This equation is only relevant for energy changes in a uniform gravitational field (such as near the Earth’s surface)

#### Derivation of GPE Equation

• When a heavy object is lifted, work is done since the object is provided with an upward force against the downward force of gravity
• Therefore energy is transferred to the object
• This equation can therefore be derived from the work done

#### Worked Example

To get to his apartment a man has to climb five flights of stairs.
The height of each flight is 3.7 m and the man has a mass of 74 kg.

What is the approximate gain in the man’s gravitational potential energy during the climb?

A.     13 000 J               B.     2700 J               C.     1500 J               D.     12 500 J

#### Exam Tip

In your exam, you will be expected to know how to derive the gravitational potential energy equation from first principles, so make sure to practice this derivation!

### Exchange Between Eₖ and Eₚ

• There are many scenarios that involve the transfer of kinetic energy into gravitational potential, or vice versa
• Some examples are:
• A swinging pendulum
• Objects in freefall
• Sports which involve falling, such as skiing and skydiving
• Using the principle of conservation of energy, and taking any drag forces as negligible:

Loss in potential energy = Gain in kinetic energy

#### Worked Example

The diagram below shows a skier on a slope descending 750 m at an angle of 25° to the horizontal.

Calculate the final speed of the skier, assuming that he starts from rest and 15% of his initial gravitational potential energy is not transferred to kinetic energy.

Step 1: Write down the known quantities

• Vertical height, h = 750 sin 25°
• Ek = 0.85 Ep

Step 2: Equate the equations for Ek and Ep

Ek = 0.85 Ep

½ mv2 = 0.85 × mgh

Step 3: Rearrange for final speed, v

Step 4: Calculate the final speed, v

#### Exam Tip

GPE:

• This equation only works for objects close to the Earth’s surface where we can consider the gravitational field to be uniform.
• At A level, you might have to consider examples where the gravitational field is not uniform, such as in space, where this equation for GPE will not be relevant.

KE:

• When using the kinetic energy equation, note that only the speed is squared, not the mass or the ½.
• If a question asks about the ‘loss of kinetic energy’, remember not to include a negative sign since energy is a scalar quantity.
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