Spring Forces
A short lesson with diagrams, a worked example, a problem to try, and a quick check. Then quiz yourself on 5 practice questions.
A spring pulls back in proportion to how far it’s stretched or compressed from its natural length: F = −kx. Springs side by side (parallel) act stiffer; springs end to end (series) act softer.
1. Combining springs
2. Springs in moving frames
If the support accelerates, use ΣF = ma: in an elevator accelerating up at a, kx − mg = ma, so the spring stretches more: x = m(g + a)/k.
On a horizontal frictionless surface, the spring force is the net force: a = −kx/m, largest at the biggest stretch.
x in F = −kx is measured from the spring’s natural length, not from where a hanging mass sits at rest. (Around that equilibrium, the net force on the mass is still −k times the displacement, which is why vertical oscillations work the same way.)
3. Example, then try it
A 0.50 kg mass hangs at rest from a spring with k = 200 N/m. How far does the spring stretch?
kx = mg → x = (0.50)(10) ÷ 200 = 0.025 m (2.5 cm).
Springs with k1 = 100 N/m and k2 = 300 N/m hold a 4.0 kg block. Find the stretch with the springs in parallel, then in series. In series, how much does each spring stretch?
Parallel: k = 100 + 300 = 400 N/m; x = 40 ÷ 400 = 0.10 m.
Series: 1/k = 1/100 + 1/300 = 4/300, so k = 75 N/m; x = 40 ÷ 75 ≈ 0.53 m.
Each spring in series carries the full 40 N: 40/100 = 0.40 m and 40/300 ≈ 0.13 m. They add to 0.53 m ✓.
Answer: 0.10 m in parallel; 0.53 m in series (0.40 m + 0.13 m).
4. Common mistakes
A block hangs from a spring inside an elevator. The elevator starts accelerating upward. How does the spring’s stretch compare with when the elevator was at rest?
kx − mg = ma, so kx = m(g + a) is larger. The spring stretches more.
Ready to practice?
5 questions on Spring Forces, each with a hint and an explanation for every wrong answer.