Gravitational potential energy

Lifting something up gives it energy in its gravitational potential store. The energy gained is ΔGPE = mgh: mass × gravitational field strength × the change in height. Twice the mass or twice the height means twice the energy, and on the Moon, where g is smaller, the same lift stores less. Falling back down, the object loses that energy again.

Key facts

  • Gravitational potential energy ΔGPE = mgh (g = 9.8 N/kg on Earth).
  • Gravitational potential energy depends on mass, height and g: on the Moon (g = 1.6 N/kg) the same lift gains far less.
  • To find a height from GPE, rearrange: h = GPE ÷ (mg).

Worked examples

Example 1

A 2 kg book is lifted 1.5 m onto a shelf. How much gravitational potential energy does it gain? (g = 10 N/kg)

  1. Multiply mass, gravitational field strength and the height it rises. E = 2 × 10 × 1.5 = 30 J

Answer: 30 J

Example 2

A 0.20 kg apple falls 2.0 m from a branch. How much gravitational potential energy does it lose? (g = 9.8 N/kg)

  1. E = m × g × h = 0.20 × 9.8 × 2.0 = 3.92 J
  2. This energy goes into the apple’s kinetic store as it speeds up.

Answer: 3.92 J

Example 3

A crane lifts a 500 kg crate 12 m. How much gravitational potential energy does the crate gain? (g = 9.8 N/kg)

  1. Use ΔGPE = mgh. ΔGPE = 500 × 9.8 × 12 = 58,800 J

Answer: 58,800 J

Example 4

Which gains more gravitational potential energy: a 2 kg bag lifted 3 m, or a 3 kg bag lifted 1.5 m? (g = 9.8 N/kg)

  1. Work out mgh for each bag. 2 × 9.8 × 3 = 58.8 J; 3 × 9.8 × 1.5 = 44.1 J
  2. So the 2 kg bag gains more.

Answer: the 2 kg bag (58.8 J, compared with 44.1 J)

Example 5

A hiker of mass 72 kg gains 67,032 J of gravitational potential energy. How high does it rise? Give your answer to 1 decimal place. (Take g = 9.8 N/kg.)

  1. Mass 72 kg, GPE = 67,032 J, g = 9.8 N/kg.
  2. Rearrange GPE = mgh to h = GPE ÷ (mg).
  3. h = 67,032 ÷ (72 × 9.8) = 95.00 m

Answer: 95 m

Example 6

A lift with passengers of mass 1,100 kg gains 1,379,840 J of gravitational potential energy. How high does it rise? Give your answer to 1 decimal place. (Take g = 9.8 N/kg.)

  1. Mass 1,100 kg, GPE = 1,379,840 J, g = 9.8 N/kg.
  2. Rearrange GPE = mgh to h = GPE ÷ (mg).
  3. h = 1,379,840 ÷ (1,100 × 9.8) = 128.00 m

Answer: 128 m