A spring with constant k is initially compressed by x from its rest length. If it is moved to twice that displacement (2x) from rest, by what factor does the stored potential energy increase?

• A. It doubles
• B. It stays the same
• C. It quadruples
• D. It halves

Answer: (C)

The main idea is that a spring's stored potential energy depends on the square of its displacement from the rest length: U = (1/2) k x^2. Doubling the displacement from x to 2x gives U' = (1/2) k (2x)^2 = (1/2) k · 4x^2 = 4 × [(1/2) k x^2]. So the energy increases by a factor of four. The other options would imply a linear or no change in energy with displacement, which isn’t how spring energy behaves.