Springs
A short lesson with diagrams, a worked example, a problem to try, and a quick check. Then quiz yourself on 4 practice questions.
A spring pulls or pushes back toward its natural length with a force proportional to how far it’s stretched or compressed: Fs = kx. It stores energy Us = ½kx². In both, x is measured from the spring’s natural length.
1. Force and stored energy
2. Measuring x
x is the change in length, not the spring’s total length. A 20 cm spring stretched to 25 cm has x = 5 cm = 0.05 m.
A stiff spring has a large k: it takes a big force to stretch it a little.
When an object hangs at rest on a spring, the spring is already stretched, so it already stores energy. At that point the spring force balances the weight: kx = mg.
3. Example, then try it
A 1.0 kg mass hangs at rest from a spring with k = 200 N/m. How far is the spring stretched, and how much energy does it store?
Balance: kx = mg → x = (1.0)(10) ÷ 200 = 0.050 m.
Us = ½(200)(0.050)² = 0.25 J.
A spring with k = 400 N/m is compressed 0.10 m and launches a 0.50 kg block across a frictionless floor. How fast does the block leave the spring? What if the spring were compressed 0.20 m instead?
Stored energy: Us = ½(400)(0.10)² = 2.0 J.
All of it becomes kinetic energy: ½(0.50)v² = 2.0 → v² = 8.0 → v ≈ 2.8 m/s.
Double the compression: Us = ½(400)(0.20)² = 8.0 J, four times as much.
New speed: ½(0.50)v² = 8.0 → v² = 32 → v ≈ 5.7 m/s, twice as fast.
Answer: about 2.8 m/s; with twice the compression, 4× the energy and 2× the speed (about 5.7 m/s).
4. Common mistakes
A spring with k = 100 N/m has a natural length of 20 cm. It is stretched to a total length of 25 cm. What force does it exert?
The stretch is x = 25 − 20 = 5 cm = 0.05 m, so F = kx = (100)(0.05) = 5.0 N. Using the total length (0.25 m) gives 25 N, which is wrong.
Ready to practice?
4 questions on Springs, each with a hint and an explanation for every wrong answer.