Hooke's Law

A spring resists being stretched or compressed, and it resists harder the farther you push it from its natural length. Pull a spring 1 cm and it tugs back gently; pull it 10 cm and it tugs back ten times as hard. That proportionality is the whole law: F = -kx, where x is the displacement from the relaxed length and k is the spring constant, a fixed number in newtons per meter that tells you how stiff the spring is. A stiff spring has a large k; a loose one has a small k.

The minus sign is not about the force being negative. It is about direction: the spring force always points opposite to the displacement, back toward equilibrium. Stretch the spring to the right and it pulls left; compress it and it pushes right. Set up a coordinate system, measure x from the relaxed length, and the sign of -kx automatically gives you the correct direction every time, whether x is positive or negative.

This restoring behavior is what makes springs the standard example of a net force that changes direction with position instead of staying constant, which is exactly what you need to get oscillation later. In this unit, Hooke's Law joins gravity, normal force, and friction as one more force you plug into Newton's second law. In a static problem, a hanging mass stretches a spring until the spring force exactly balances gravity, so kx = mg; in a dynamic problem, F = -kx becomes the net force driving acceleration, and that equation is the seed of simple harmonic motion.