Conservative forces and potential energy

Work that comes back

Lift a book onto a shelf and you do work against gravity. Let it fall, and gravity gives the work back as kinetic energy. Push a box across a rough floor, though, and the work is gone: friction turns it into heat, and pushing the box back costs you all over again.

Conservative forces

A force is conservative if the work it does on a body moving from A to B is the same whatever path the body takes. Then the work round any closed loop is zero.

• Gravity and the force of a spring are conservative.
• Friction and air drag are not: the longer the path, the more work they take.

Explore it with ramps of the same height.

You should find that gravity's work stays $mgh$ for every angle, while friction takes more on a gentle, longer ramp.

Potential energy

For a conservative force we can keep an account: the potential energy $U$, so that the force's work equals the drop in $U$:

Near the Earth, gravity gives $U = mgh$, with $h$ measured from any level we choose; only changes in $U$ matter. A spring gives $U = \tfrac12 kx^2$.

Force from potential energy

Run the account backwards: the force is the downhill slope of $U$,

Where $U$ has a minimum, $F = 0$ and the body sits in stable equilibrium, like a marble in a bowl.

Watch out. Potential energy belongs to a body and the Earth (or the spring) together. And the minus sign matters: the force points toward lower $U$.

A worked example

A $62\ \text{kg}$ hiker climbs $350\ \text{m}$ up a mountain, once by the steep path and once by the long winding road.

1. Gravity's work. On either route, $W = -mgh = -62 \times 9.8\ \text{m/s}^2 \times 350\ \text{m} = -2.13 \times 10^{5}\ \text{J}$.
2. Potential energy. It rises by $2.13 \times 10^{5}\ \text{J}$ either way.
3. What differs. The winding road takes longer and tires different muscles, but gravity does not care about the path.

Think first. If a force's work depends on the path, it has no potential energy. Then you must work out its work along the actual path.

Practise with new numbers

Try it in Python

The program slides a block down four ramps of the same height and compares gravity's work with friction's.

Remember these

• Conservative force: its work depends only on the start and the end.
• $W = -\Delta U$; near the Earth $U = mgh$; a spring $U = \tfrac12 kx^2$.
• $F = -\dfrac{dU}{dx}$; a minimum of $U$ is a stable equilibrium.

Check yourself

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

1. Is the force of a stretched spring conservative? And friction from a floor?

2. What is the potential energy of a $0.50\ \text{kg}$ ball $12\ \text{m}$ above the ground, taking $U = 0$ at the ground? ($g = 9.8\ \text{m/s}^2$.)

3. A particle has $U = 5x^2$ joules. What is the force on it at $x = 0.40\ \text{m}$?

4. A ball goes up a hill and comes back down, on a frictionless track. How much work did gravity do on the round trip?