The scalar product and the work done by a constant force

Work in everyday words and in physics

Holding a heavy bag still for an hour makes your arms ache. Yet in physics you do no work on the bag, because it does not move. In physics, work needs two things: a force, and a displacement along it.

The scalar product

Two vectors can be multiplied to give a plain number, the scalar product (or dot product):

where $\theta$ is the angle between them. It is $B\cos\theta$, the part of $\vec{B}$ along $\vec{A}$, times $A$.

From the parts of the vectors, it is easier still:

since $\hat{i} \cdot \hat{i} = 1$ and $\hat{i} \cdot \hat{j} = 0$.

The work of a steady force

A steady force $\vec{F}$ acts on a body while it moves through $\vec{d}$. The work done by the force is

Work is a scalar, measured in joules: $1\ \text{J} = 1\ \text{N m}$.

The angle decides the sign. Try it.

You should find that the work is largest at $0^\circ$, zero at $90^\circ$, and negative beyond it, down to $-Fd$ at $180^\circ$.

Watch out. "Zero work" does not mean "no force". The normal reaction on a sliding box and the pull of a string on a whirling stone are real forces, but they act at right angles to the motion, so they do no work.

A worked example

Rohan pushes a lawn mower $18\ \text{m}$ across a lawn. He pushes along the handle with $85\ \text{N}$, at $40^\circ$ below the horizontal.

1. The angle with the motion. The mower moves along the ground, so $\theta = 40^\circ$.
2. Work by the push. $W = 85\ \text{N} \times 18\ \text{m} \times \cos 40^\circ = 1172\ \text{J}$.
3. Other forces. The mower's weight and the ground's push are vertical, at $90^\circ$ to the motion: they do no work.

Think first. Before any formula, ask which way the force points and which way the body moves. The angle between them is the whole story.

Practise with new numbers

Try it in Python

The program works out the work of one pull at seven angles.

Remember these

• $\vec{A} \cdot \vec{B} = AB\cos\theta = A_xB_x + A_yB_y + A_zB_z$, a plain number.
• Work by a steady force: $W = Fd\cos\theta$, in joules.
• Positive when the force helps the motion, zero at $90^\circ$, negative when it opposes it.

Check yourself

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

1. A waiter carries a tray across a room at a steady height. How much work does his upward push do on the tray?

2. A pull of $50\ \text{N}$ at $60^\circ$ to the motion acts over $8.0\ \text{m}$. How much work does it do?

3. A force $\vec{F} = (4\hat{i} - 3\hat{j})\ \text{N}$ moves a body through $\vec{d} = (2\hat{i} + 5\hat{j})\ \text{m}$. Find the work. What does its sign tell you?

4. A satellite moves in a circular orbit. How much work does gravity do on it in one orbit?