Elastic Energy

Compression and Extension It takes force to press a spring together. More compression requires stronger force. It takes force to extend a spring. More extension requires stronger force.

Spring Constant The distance a spring moves is proportional to the force applied. The ratio of the force to the distance is the spring constant (k). F x

Hooke’s Law The force from the spring attempts to restore the original length. This is sometimes called Hooke’s law. The distance x is the displacement from the natural length, L. L L+ x L - x

Scales One common use for a spring is to measure weight. The displacement of the spring measures the mass. Fs = -k(-y) -y Fg = -mg

Stiff Springs Two spring scales measure the same mass, 200 g. One stretches 8.0 cm and the other stretches 1.0 cm. What are the spring constants for the two springs? The spring force balances the force from gravity: F = 0 = (-mg) + (-kx). Solve for k = mg/ (–x). x is negative. Substitute values: (0.20 kg)(9.8 m/s2)/(0.080 m) = 25 N/m. (0.20 kg)(9.8 m/s2)/(0.010 m) = 2.0 x 102 N/m.

Force and Distance The force applied to a spring increases as the distance increases. The product within a small step is the area of a rectangle (kx) Dx. The total equals the area between the curve and the x axis. F F = kx x Dx

Work on a Spring For the spring force the force makes a straight line. The area under the line is the area of a triangle. F Fs=kx x x

Elastic Work The elastic force exerted by a spring becomes work. W = -(1/2) kx2 The work done by the spring as it compresses is negative. Like gravity the path taken to the end doesn’t matter. Spring force is conservative Potential energy, U = (1/2)kx2 Ws = -(1/2)ky2 Fs = -k(-y) -y

Springs and Conservation The spring force is conservative. U = ½ kx2 The total energy is E = ½ mv2 + ½ kx2 A 35 metric ton box car moving at 7.5 m/s is brought to a stop by a bumper. The bumper has a spring constant of 2.8 MN/m. Initially, there is no bumper E = ½ mv2 = 980 kJ Afterward, there is no speed E = ½ kx2 = 980 kJ x = 0.84 m v x

Energy Conversion A 30 kg child pushes down 15 cm on a trampoline and is launched 1.2 m in the air. What is the spring constant? Initially the energy is in the trampoline. U = ½ ky2 Then the child has all kinetic energy, which becomes gravitational energy. U = mgh The energy is conserved. ½ ky2 = mgh k = 3.1 x 104 N/m