Gravitational potential energy

When objects are above the ground, they have the potential to fall down to the ground.

When they do fall, they transfer that gravitational potential energy into kinetic energy as they speed up.

Calculating the change in gravitational potential energy

If we know the:

mass of the object, in kilograms, kg
height the object is lifted up by, in metres, m
gravitational field strength, in newtons per kilogram, N/kg

…then we can calculate the change in gravitational potential energy when the object is lifted up or falls down!

The formula we use is:

\text{Change in gravitational potential energy} = \text{mass} \times \text{gravitational field strength} \times \text{change in height above ground}

Putting it into symbols, we get:

\Delta E_p = mg\Delta h

Where:

\Delta E_p is the change in gravitational potential energy, in joules, J
m is the mass of the object, in kilograms, kg
g is the gravitational field strength, in newtons per kilogram, N/kg
\Delta h is the change in the object’s distance from the ground, in metres, m

Height

Make sure the height is the shortest distance from the object to the ground.

The height should be perpendicular to the gravitational field lines (or parallel to the force of gravity).

‘Change in’?

We can’t directly calculate the gravitational potential energy of an object, but we can calculate its change in gravitational potential energy when it falls or is lifted up.

That’s why we put the \Delta symbol in front of E_p and h in the formula, to show that we’re calculating the change in gravitational potential energy and height, not the actual gravitational potential energy or height.

If we were calculating the actual gravitational potential energy, that would suggest that an object on the surface of the Earth has zero gravitational potential energy, which isn’t true - the surface of the Earth isn’t the absolute point where gravitational potential energy is zero, it’s just the point where we usually choose to measure it from.

Examples

A 2 kg object is lifted up by 5 m. The gravitational field strength is 10 N/kg. What is the change in gravitational potential energy of the object?

m = 2\,kg
\Delta h = 5\,m
g = 10\,N/kg
\Delta E_p = mg\Delta h
\Delta E_p = 2\,kg \times 10\,N/kg \times 5\,m
\Delta E_p = 100\,J

The gravitational field strength on the Moon is 1.6 N/kg. A 3 kg object is lifted up by 4 m on the Moon. What is the change in gravitational potential energy of the object?

m = 3\,kg
\Delta h = 4\,m
g = 1.6\,N/kg
\Delta E_p = mg\Delta h
\Delta E_p = 3\,kg \times 1.6\,N/kg \times 4\,m
\Delta E_p = 19.2\,J

An object gains 49 J of gravitational potential energy when it is lifted up by 10 m. The gravitational field strength is 9.8 N/kg. What is the mass of the object?

\Delta E_p = 49\,J
\Delta h = 10\,m
g = 9.8\,N/kg
\Delta E_p = mg\Delta h
m = \frac{\Delta E_p}{g\Delta h}
m = \frac{49\,J}{9.8\,N/kg \times 10\,m}
m = 0.5\,kg

An object loses 20 J of gravitational potential energy when it falls down by 2 m. The gravitational field strength is 10 N/kg. What is the mass of the object?

\Delta E_p = -20\,J (the object loses energy, so the change in energy is negative)
\Delta h = -2\,m (the object falls down, so the change in height is negative)
g = 10\,N/kg
\Delta E_p = mg\Delta h
m = \frac{\Delta E_p}{g\Delta h}
m = \frac{-20\,J}{10\,N/kg \times -2\,m}
m = 1\,kg

How far does a 2 kg object fall if it loses 80 J of gravitational potential energy? Assume the gravitational field strength is 10 N/kg.

m = 2\,kg
\Delta E_p = -80\,J (the object loses energy, so the change in energy is negative)
g = 10\,N/kg
\Delta E_p = mg\Delta h
\Delta h = \frac{\Delta E_p}{mg}
\Delta h = \frac{-80\,J}{2\,kg \times 10\,N/kg}
\Delta h = -4\,m
So the object falls down by 4 m.

A 5 kg object is lifted up on a far-away mega planet, where the gravitational field strength is 80 N/kg. The object gains 200 J of gravitational potential energy. How far is the object lifted up?

m = 5\,kg
\Delta E_p = 200\,J
g = 80\,N/kg
\Delta E_p = mg\Delta h
\Delta h = \frac{\Delta E_p}{mg}
\Delta h = \frac{200\,J}{5\,kg \times 80\,N/kg}
\Delta h = 0.5\,m

A 5 kg object is lifted up by 3 m on the Moon, where the gravitational field strength is 1.6 N/kg. How much gravitational potential energy does the object gain?

m = 5\,kg
\Delta h = 3\,m
g = 1.6\,N/kg
\Delta E_p = mg\Delta h
\Delta E_p = 5\,kg \times 1.6\,N/kg \times 3\,m
\Delta E_p = 24\,J

flashcards

Question	Answer
What is gravitational potential energy?	Energy stored in an object due to its position above the ground, giving it the potential to fall down.
What two forms of energy are involved when an object falls?	Gravitational potential energy is transferred into kinetic energy as the object speeds up.
What three quantities are needed to calculate the change in gravitational potential energy?	Mass (in kg), height change (in m), and gravitational field strength (in N/kg).
State the formula for change in gravitational potential energy in words.	Change in gravitational potential energy = mass × gravitational field strength × change in height above ground.
State the formula for change in gravitational potential energy in symbols.	\Delta E_p = mg\Delta h
What does \Delta E_p represent?	The change in gravitational potential energy, measured in joules (J).
What does m represent in the formula \Delta E_p = mg\Delta h?	The mass of the object, in kilograms (kg).
What does g represent in the formula \Delta E_p = mg\Delta h?	The gravitational field strength, in newtons per kilogram (N/kg).
What does \Delta h represent in the formula \Delta E_p = mg\Delta h?	The change in the object’s distance from the ground, in metres (m).
How should height be measured when calculating gravitational potential energy?	The height should be the shortest distance from the object to the ground, perpendicular to the gravitational field lines (parallel to the force of gravity).
Why do we use ‘change in’ (\Delta) for gravitational potential energy?	We cannot directly calculate the actual gravitational potential energy of an object, only the change when it falls or is lifted.
Why is an object on the Earth’s surface not considered to have zero gravitational potential energy?	The Earth’s surface is not the absolute zero point for gravitational potential energy; it is just the point we choose to measure from.
Calculate: A 2 kg object is lifted by 5 m where g = 10 N/kg. What is \Delta E_p?	\Delta E_p = 2 \times 10 \times 5 = 100 J
Calculate: On the Moon (g = 1.6 N/kg), a 3 kg object is lifted 4 m. What is \Delta E_p?	\Delta E_p = 3 \times 1.6 \times 4 = 19.2 J
Calculate: An object gains 49 J of GPE when lifted 10 m (g = 9.8 N/kg). What is its mass?	m = \frac{49}{9.8 \times 10} = 0.5 kg
How do you handle signs when an object falls down and loses gravitational potential energy?	The change in energy (\Delta E_p) is negative (e.g., -20 J) and the change in height (\Delta h) is negative (e.g., -2 m), resulting in a positive mass.
Calculate: An object loses 20 J falling 2 m (g = 10 N/kg). What is its mass?	m = \frac{-20}{10 \times -2} = 1 kg
Calculate: A 2 kg object loses 80 J of GPE (g = 10 N/kg). How far does it fall?	\Delta h = \frac{-80}{2 \times 10} = -4 m, so it falls 4 m.
Calculate: On a mega planet (g = 80 N/kg), a 5 kg object gains 200 J. How far is it lifted?	\Delta h = \frac{200}{5 \times 80} = 0.5 m
Calculate: A 5 kg object is lifted 3 m on the Moon (g = 1.6 N/kg). How much GPE does it gain?	\Delta E_p = 5 \times 1.6 \times 3 = 24 J