Hooke's law

Meaning: A spring displaced from its equilibrium length pushes back toward equilibrium with a force proportional to the displacement. A graph of force versus displacement is a straight line through the origin, and the slope's magnitude is k, the stiffness. The law holds only within the proportional (elastic) limit; beyond it the relationship is no longer linear.

Equation: F = −kx, where F is the restoring force (N), k is the spring constant (N/m), and x is the displacement from equilibrium (m). Stored energy is U = ½kx², with U in joules (J).

On the MCAT: A mass hanging at rest on a spring gives kx = mg. Force-extension graphs: the slope gives k and the area under the line gives the stored energy. In simple harmonic motion the force is greatest at maximum displacement and zero at equilibrium. The period is T = 2π√(m/k), with T in seconds and m in kg, so a stiffer spring gives a shorter period.

Common confusion: x is the change from the relaxed length, not the spring's total length. The minus sign shows that the force opposes the displacement; it does not make the magnitude negative. Force is linear in x but energy goes as x², so doubling the stretch doubles the force and quadruples the energy.