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Weight is a force, W = mg

Forces and motion · level 5, lesson 4 of 6

A question to start

Drop a hammer and a feather on the Moon, where there is no air. Which lands first? An astronaut tried it in 1971. What happened, and what does it say about weight and mass?

The idea

The weight of an object is the force of gravity on it:

\[W = m g,\]

where \(g\) is the gravitational field strength at that place, in newtons per kilogram. It is the same number as the free-fall acceleration in m/s², because \(1 \text{ N/kg} = 1 \text{ m/s}^2\).

Mars Earth
\(g\) at the surface 3.71 N/kg 9.81 N/kg
Weight of 1 kg 3.71 N 9.81 N
Weight of a 1,025 kg rover 3,800 N 10,060 N
Weight of an 80 kg person 297 N 785 N

Now put weight into the second law for a thing falling with nothing else acting:

\[a = \frac{F}{m} = \frac{mg}{m} = g.\]

The mass cancels. A heavy thing has a bigger pull, but also more inertia, by exactly the same factor. So everything falls with the same acceleration, \(g\), if air does not get in the way. On Apollo 15, David Scott dropped a hammer and a feather together on the Moon. They landed at the same moment.

W = 5mgW = mga = ga = g

Worked examples

1. A sample bag. A 2.5 kg bag of rocks on Mars: \(W = 2.5 \times 3.71 = 9.3\) N. On Earth, \(24.5\) N.

2. The pull needed to lift it at a steady speed. Balanced forces: the lift equals the weight, 9.3 N on Mars.

3. Lifting it with an acceleration. To lift the same bag upward at 1 m/s² on Mars: \(F - mg = ma\), so \(F = m(g + a) = 2.5 \times 4.71 = 11.8\) N.

Try it

  1. Find the weight of a 50 kg crate on Mars and on Earth.
  2. A spring scale on Mars reads 74.2 N for a box. What is its mass?
  3. Why does a heavier thing not fall faster, if air does not get in the way?
  4. What upward force lifts a 10 kg box at 0.5 m/s² upward acceleration on Mars?
Answers
  1. Mars \(50 \times 3.71 = 186\) N; Earth \(50 \times 9.81 = 491\) N.
  2. \(74.2 \div 3.71 = 20\) kg.
  3. Its bigger weight is matched by its bigger mass: \(a = mg/m = g\) for every object.
  4. \(F = 10 \times (3.71 + 0.5) = 42.1\) N.

Common mistake

"\(g\) is the same everywhere because it is a constant." Little \(g\) changes from place to place: 9.81 on Earth, 3.71 on Mars, 1.62 on the Moon. It is big \(G\), in Newton's law of gravity, that is the same everywhere.

Where it leads