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Gravitational potential energy

Energy and power · level 5, lesson 4 of 6

A question to start

A water tank sits on a tower 20 m above a Mars habitat. On Earth, a tank that high gives good water pressure for free. Does it store as much energy on Mars?

The idea

Lift a mass \(m\) by a height \(h\) at a steady speed. The force needed equals its weight \(mg\), and it moves a distance \(h\), so the work done is \(mgh\). That work is stored as gravitational potential energy:

\[E_p = mgh.\]

It is "potential" because it can come back: let the mass fall, and the stored energy turns into kinetic energy.

Only changes in height matter. You can measure \(h\) from any level you like, the floor, the ground or the bottom of a crater, as long as you are consistent.

Lifting 1,000 kg (a cubic metre of water) by 20 m Mars Earth
\(g\) 3.71 N/kg 9.81 N/kg
Energy stored, \(mgh\) 74 kJ 196 kJ
Water pressure at the bottom of the pipe, \(\rho g h\) 74 kPa 196 kPa

So on Mars, the tank stores 0.38 as much energy and gives 0.38 of the pressure. To get Earth's pressure from a tank, you would need a tower 2.6 times as tall, or a pump.

hhabitat

The formula \(mgh\) is only right where \(g\) is about the same over the whole height: near a planet's surface. For rockets and orbits, gravity weakens with height, and a different formula is needed.

Worked examples

1. The water tank. \(1000 \times 3.71 \times 20 = 74{,}200\) J on Mars; \(196{,}200\) J on Earth.

2. A climber on Olympus Mons. The volcano's summit is about 22 km above the Mars datum, the planet's "sea level". Climbing from the datum with a 150 kg suit and pack: \(150 \times 3.71 \times 22{,}000 = 1.22\times10^7\) J, about 3.4 kWh. (Gravity is nearly the same at the top: 22 km is small compared with the radius of Mars.)

3. Choosing the zero. A book on a shelf 2 m up in a room on Earth, the floor 3 m above the ground. Relative to the floor: \(1 \times 9.81 \times 2 = 19.6\) J. Relative to the ground: 49 J. Only differences matter.

Try it

  1. Find the potential energy gained by a 5 kg rock lifted 2 m on Mars.
  2. How much energy is stored lifting a 300 kg module by 12 m on Earth?
  3. A pump lifts 2,000 kg of water up 15 m on Mars. What energy does it give the water?
  4. Why can't \(mgh\) be used for a rocket climbing to 1,000 km?
Answers
  1. \(5 \times 3.71 \times 2 = 37.1\) J.
  2. \(300 \times 9.81 \times 12 = 35{,}300\) J.
  3. \(2000 \times 3.71 \times 15 = 111{,}300\) J.
  4. Gravity is noticeably weaker that high, so \(g\) is not constant over the climb.

Common mistake

Using the length of the path instead of the height. Walking 100 m up a gentle ramp that rises 5 m gains \(mg \times 5\), not \(mg \times 100\). Only the change in height counts.

Where it leads