How stretch changes stored energy

A ball-launcher toy fires with a spring. Yael compresses the spring 0.10 m for a soft shot, and 0.20 m for a hard one.

Twice the compression. Does that store twice the energy, or more?

The toy's spring has k = 500 N/m. At 0.10 m: x² is 0.10 × 0.10 = 0.01, and EPE = (½ × 500 × 0.10) × 0.10 = 2.5, so 2.5 J.

At 0.20 m: x² is 0.20 × 0.20 = 0.04, and EPE = (½ × 500 × 0.20) × 0.20 = 10, so 10 J.

Twice the compression stored four times the energy: 10 J against 2.5 J.

The reason sits in the squared x. Doubling x doubles the force needed, and it also doubles the distance that force acts through; the two doublings multiply.

Doubling x doubles the base and the height: the triangle's area quadruples.

Here is the rule. With the spring unchanged, multiplying the extension by a number multiplies the stored energy by that number squared.

Triple the extension: nine times the energy. Halve it: a quarter.

The energy factor is the extension factor, squared.

At 0.30 m: x² is 0.30 × 0.30 = 0.09, and EPE = (½ × 500 × 0.30) × 0.30 = 22.5, so 22.5 J, nine times the soft shot's 2.5 J.

x² carries the growth: ×2 in stretch is ×4 in energy; ×3 is ×9.

A notebook clip's spring stores some energy at 0.02 m of extension. The extension is doubled to 0.04 m. What happens to the stored energy?

Doubling a spring's extension quadruples its stored energy. Why?

A jump-counter's spring stores 1.5 J at 0.05 m of extension. The same spring is stretched to 0.15 m. How much energy does it store?

The extension tripled, from 0.05 m to 0.15 m.

The stored energy multiplies by 3 squared, which is 9.

EPE = 1.5 × 9 = 13.5

The spring stores 13.5 J.

A gate spring stores 8.0 J at 0.40 m of extension. The stretch is eased back to 0.20 m. How much energy remains stored?

The extension halved.

The stored energy multiplies by a half squared: a quarter.

EPE = 8.0 ÷ 4 = 2.0

The spring now stores 2.0 J.

A screen-door spring stores 0.60 J at its usual extension. A stronger door stop triples that extension. Enter the energy stored at the tripled extension, in J.

Your turn

A mousetrap's spring stores energy at 0.010 m of compression. Setting the trap compresses it to 0.020 m instead. What happens to the stored energy?

A pinball plunger's spring stores 1.6 J at full draw. Checked mid-game, it is storing 0.4 J. Compared with the full draw, its extension now is...

A camping-chair spring stores 0.80 J at 0.05 m of extension. Enter the energy it stores at 0.10 m of extension, in J.