Free-body diagrams, equilibrium and the lift

A drawing that solves the problem

Almost every mistake in mechanics starts with a force left out or a force put in that is not there. The cure is a free-body diagram:

1. Draw the body alone, cut free from everything around it.
2. Draw an arrow for every force on it: weight, and one force for each thing it touches (a normal reaction, friction, a tension, a push).
3. Choose axes, often along and across the motion.
4. Apply $F_{\text{net}} = ma$ along each axis.

Watch out. Do not draw the forces the body exerts on others. There is also no "force of motion". A body moving to the right needs no force to the right, unless it is speeding up.

Equilibrium of a particle

If the net force is zero, the body is in equilibrium: at rest, or moving at a steady velocity. Then the forces balance along every axis:

For three forces in equilibrium, the arrows drawn head to tail make a closed triangle.

A person in a lift

You stand on a scale in a lift. Two forces act on you: your weight $mg$ down, and the scale's push $R$ up. The scale shows $R$. Taking up as positive:

What matters is the acceleration, not the direction of motion. Play with it.

You should find that a lift moving at a steady speed, up or down, gives your normal weight, and a lift in free fall ($a = -9.8$) gives zero: you feel weightless.

A worked example

A traffic light of mass $15\ \text{kg}$ hangs from two equal cables, each at $35^\circ$ to the horizontal. Find the tension.

1. Free-body diagram. The weight $147\ \text{N}$ down, and two tensions along the cables.
2. Horizontally the two horizontal parts cancel, since the cables are equal.
3. Vertically. Each cable lifts $T\sin 35^\circ$: $2T\sin 35^\circ = 147\ \text{N}$, so $T = \dfrac{147\ \text{N}}{2 \times 0.574} = 128\ \text{N}$.

Think first. Each cable pulls with $128\ \text{N}$, almost the whole weight, because flat cables have only a small upward part. A cable that is nearly horizontal needs an enormous tension.

Practise with new numbers

Try it in Python

The program follows a girl on a scale through a whole lift ride.

Remember these

• Free-body diagram: one body, every force on it, nothing else.
• Equilibrium: $\Sigma F_x = 0$ and $\Sigma F_y = 0$.
• In a lift, $R = m(g + a)$ with up positive: the acceleration decides, not the direction of motion.
• Free fall: the scale reads zero.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. A ball is thrown straight up. Draw (in words) its free-body diagram at the top of its path, ignoring air.

2. A $65\ \text{kg}$ man is in a lift that slows down while going down, at $1.5\ \text{m/s}^2$. What does the scale read? ($g = 9.8\ \text{m/s}^2$.)

3. A lamp hangs at rest from one string. What is the tension, for a $2.0\ \text{kg}$ lamp?

4. Why can a nearly horizontal washing line snap under a light shirt?