← Mechanics

Springs and elastic potential energy

A spring-based scenario (a kitchen scale, a bungee, a trampoline) has appeared regularly, and needs its own formula pair beyond the standard energy toolkit.

Hooke's law and elastic potential energy

  • Hooke's law: F = kx, where k is the spring constant (N m⁻¹) and x is the extension or compression from the natural length.
  • Elastic potential energy stored: E_e = ½kx² — analogous in shape to E_k = ½mv², both built from a squared term times a constant.
  • A stiffer spring has a LARGER k, meaning more force is needed for the same extension, and more energy is stored for the same extension too.
Worked example Finding a spring constant from a measurement 0.11 kg of flour rests on a kitchen scale supported by 3 identical springs, each compressing 0.01 m. Find the spring constant of one spring. The total weight is shared equally by the 3 identical springs: total force = mg = 0.11 × 9.8 = 1.078 N, so each spring supports 1.078/3 ≈ 0.359 N. Apply Hooke's law to one spring: F = kx, so k = F/x = 0.359 / 0.01 = 35.9 N m⁻¹. ✓ k ≈ 36 N m⁻¹ per spring. Dividing the total force by the number of identical springs BEFORE applying F = kx is the step most often skipped. ✕

F = kx and E_k = ½mv² are not interchangeable with E_e = ½kx²

A spring problem needs the ELASTIC potential energy formula E_e = ½kx², built from the spring constant and extension — not the kinetic energy formula ½mv², even though both have the same "half times a squared quantity" shape. Mixing them up (e.g. using ½kv² or ½mx²) produces an answer with the wrong units entirely, which is usually the fastest way to catch the error yourself. ℹ

Energy conservation with a spring

When something compresses or stretches a spring and comes to rest (a tackle bag absorbing a player, a bungee cord at full stretch), the kinetic energy just before contact converts entirely into elastic potential energy at the point of maximum compression: ½mv² = ½kx². This single equation is the bridge between the mechanics toolkit (KE) and the spring toolkit (elastic PE), and is exactly what "reasoning about a change without calculating" questions in this area are built around.

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