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Free-body diagrams

Forces and motion · level 5, lesson 6 of 6

A question to start

An astronaut stands on a bathroom scale in a lift on Mars. The lift starts moving up. Does the scale read more, less, or the same? Guessing is hard. Is there a sure way to work it out?

The idea

A free-body diagram shows one object on its own, with an arrow for every force on it, and nothing else. It turns a force problem into arithmetic. The steps:

  1. Pick one object. Draw it as a box or a dot.
  2. Draw every force acting on it: weight, normal force, friction, tension, thrust, drag. Not the forces it exerts on other things.
  3. Choose a positive direction, usually the way it accelerates.
  4. Write \(F_\text{net} = ma\) along that direction and solve.

For the astronaut in the lift: weight \(mg\) down, and the scale's push \(N\) up. The scale reads \(N\).

Nmga

With up as positive: \(N - mg = ma\), so

\[N = m(g + a).\]

Going up and speeding up, the scale reads more than usual. Slowing down on the way up (\(a\) negative), it reads less.

Worked examples

1. The lift on Mars. Astronaut and suit, 100 kg. The lift accelerates upward at 1.0 m/s². \(N = 100 \times (3.71 + 1.0) = 471\) N, compared with 371 N standing still.

2. A crate dragged by a rope. A 40 kg crate on a flat Mars floor is pulled with 60 N; friction is 25 N. Horizontal: \(60 - 25 = 40a\), so \(a = 0.875\) m/s². Vertical: \(N = mg = 40 \times 3.71 = 148\) N (no vertical acceleration).

3. A lander slowing down. A 1,000 kg lander falls while its engines push up with 5,000 N. Down is the way it moves; take up as positive: \(5000 - 1000 \times 3.71 = 1000 a\), so \(a = +1.29\) m/s², upward. It is slowing its fall.

Try it

  1. What goes into a free-body diagram, and what is left out?
  2. A 70 kg person in a lift on Earth accelerating up at 2 m/s²: what does the scale read in newtons?
  3. The same lift on Mars: what does it read?
  4. A 1,000 kg lander on Mars with a thrust of 3,000 N. Is it speeding up or slowing down as it falls?
Answers
  1. In: every force acting on the one chosen object. Out: other objects, and forces the object exerts on them.
  2. \(70 \times (9.81 + 2) = 827\) N.
  3. \(70 \times (3.71 + 2) = 400\) N.
  4. \(3000 - 3710 = -710\) N, a net force downward: it is still speeding up as it falls.

Common mistake

Drawing forces the object exerts. The astronaut pushes down on the scale, but that force acts on the scale, not the astronaut. Only forces on the chosen object go in its diagram.

Where it leads