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Forces on a ramp

Forces and motion · level 6, lesson 3 of 6

A question to start

A cart of rock samples sits on a ramp sloping at 20° into a Mars habitat. How hard does gravity pull it down the ramp? How hard does it press into the ramp?

The idea

Weight \(mg\) points straight down. On a slope, it is useful to split it into two parts at right angles:

  • Along the slope (down it): \(mg \sin\theta\).
  • Into the slope (perpendicular to it): \(mg \cos\theta\).

Here \(\theta\) is the angle of the slope. These two parts together make up the whole weight.

θmgmg sin θmg cos θN

Into the slope, nothing moves, so the ramp pushes back with the normal force

\[N = mg\cos\theta.\]

That is less than \(mg\): on a slope, an object presses less hard on the surface than on flat ground. So the friction limit \(\mu_s N\) is smaller too.

Check the extremes: on flat ground (\(\theta = 0\)), \(\sin 0 = 0\) and \(\cos 0 = 1\), so no pull along it and \(N = mg\). On a vertical wall (\(\theta = 90°\)), all of the weight pulls along the "slope", and \(N = 0\).

Worked examples

1. The sample cart on Mars. \(m = 150\) kg, \(\theta = 20°\). Weight \(= 150 \times 3.71 = 557\) N. Along the slope: \(557 \times \sin 20° = 557 \times 0.342 = 190\) N. Into it: \(557 \times 0.940 = 523\) N.

2. The same cart on Earth. Weight \(1{,}472\) N. Along: \(503\) N. Into: \(1{,}383\) N. Both parts are 2.6 times bigger, the ratio of the two \(g\) values.

3. Holding it still. With no friction, a rope up the slope must pull with 190 N on Mars to hold the cart in place.

Try it

  1. Write the part of the weight along a slope of angle \(\theta\).
  2. A 40 kg box rests on a 30° slope on Mars. Find the part of its weight down the slope.
  3. Find the normal force on it.
  4. What happens to \(N\) as the slope gets steeper?
Answers
  1. \(mg\sin\theta\).
  2. \(40 \times 3.71 \times 0.5 = 74.2\) N.
  3. \(40 \times 3.71 \times 0.866 = 128.5\) N.
  4. It gets smaller, as \(\cos\theta\) falls.

Common mistake

Swapping sine and cosine. A quick check: on a gentle slope, the part along it should be small. \(\sin\theta\) is small for small \(\theta\), so it belongs to the part along the slope.

Where it leads