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). x F

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+ x L - x L

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

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/s 2 )/(0.080 m) = 25 N/m.  (0.20 kg)(9.8 m/s 2 )/(0.010 m) = 2.0 x 10 2 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)  x.  The total equals the area between the curve and the x axis. F xx x F = kx

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 s =kx x x F

Elastic Work  The elastic force exerted by a spring becomes work. W =  (1/2) kx 2W =  (1/2) kx 2 The work done by the spring as it compresses is negative.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 conservativeSpring force is conservative Potential energy, U = (1/2)kx 2Potential energy, U = (1/2)kx 2 -y W s =  ky 2 F s = -k(-y)

Springs and Conservation  The spring force is conservative. U = ½ kx 2U = ½ kx 2  The total energy is E = ½ mv 2 + ½ kx 2E = ½ mv 2 + ½ kx 2  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 mv 2 = 980 kJ Afterward, there is no speed E = ½ kx 2 = 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 ky 2  Then the child has all kinetic energy, which becomes gravitational energy. U = mgh  The energy is conserved. ½ ky 2 = mgh k = 3.1 x 10 4 N/m