Plain-English Meaning
Stretch a spring twice as far and it pulls back twice as hard. The minus sign means the spring always pulls back toward its natural length — it's a restoring force. k is the spring constant (measured in N/m); stiffer springs have larger k.
Mechanics questions usually become easier once you identify whether the problem is about force balance, kinematics, energy, or conservation.
Deeper Explanation
Hooke's law defines simple harmonic motion (SHM): combining F = −kx with F = ma gives ẍ = −(k/m)x, with solution x(t) = A cos(ω t + φ) where ω = √(k/m). Elastic PE stored: U = ½kx².
Worked Example
Problem: A spring with k = 400 N/m is compressed 8 cm. What is the restoring force and elastic PE?
- F = −kx = −400 × (−0.08) = +32 N (restoring, so directed toward equilibrium).
- PE = ½kx² = ½ × 400 × (0.08)² = ½ × 400 × 0.0064 = 1.28 J.
Result: F = 32 N; PE = 1.28 J
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At A Glance
Category: Mechanics
Levels covered: High School, College, Masters, PhD
Best use: Start with the formula meaning, then move to the worked example and quiz so the equation turns into a tool instead of a memorised line.