Stretch a rubber band and it snaps back. Compress a spring and it bounces back. Bend a ruler slightly and it straightens when released. These are examples of elastic behavior, where a material returns to its original shape after the deforming force is removed. Hooke’s law states that for many materials, the force needed is proportional to the displacement. F equals negative kx, where k is the spring constant and x is the displacement from equilibrium.
The spring constant k measures stiffness. A stiff spring has a large k and requires more force to stretch. A weak spring has a small k. Spring constants vary enormously. The suspension springs in a car might have k values of about 20,000 N/m. A ballpoint pen spring might be around 200 N/m. The spring in a watch balance wheel might be less than 1 N/m.
The negative sign in Hooke’s law indicates a restoring force. If you pull the spring to the right, the force pulls it back to the left. If you compress it to the left, the force pushes it right. This restoring force is what causes oscillations. Release a compressed spring, and it overshoots equilibrium, stretches, comes back, and oscillates. The period of oscillation depends only on the mass and the spring constant: T equals 2 pi times the square root of m over k.
Elasticity has limits. Every material has an elastic limit beyond which it deforms permanently. Stretch a paper clip slightly and it bounces back. Bend it far enough and it stays bent. The stress-strain curve shows this transition. At low stresses, the relationship is linear (Hooke’s law region). Beyond the yield point, permanent deformation occurs. Eventually, the material reaches its ultimate tensile strength and breaks.
Young’s modulus is a material property that describes stiffness in tension or compression. Steel has a Young’s modulus of about 200 GPa. Rubber is about 0.01 to 0.1 GPa. Concrete is about 30 GPa. Higher values mean stiffer materials. A steel beam deflects less than a wood beam under the same load because steel is much stiffer.
Energy stored in a spring is one half kx squared. This is elastic potential energy, and it is the basis of many mechanical devices. Mouse traps store energy when you set them and release it when triggered. Archery bows store energy as you draw them and transfer it to the arrow. Shock absorbers in cars use springs (along with dampers) to absorb energy from bumps.