Consider a spring of mass M, equilibrium length L0, and spring constant k. The work done to stretch or compress the spring to a distance of L is 1/2 kX^2, where X = L − L0. Consider a spring as described above that has one end fixed and the other end moving with speed v. Assume that the speed of points along the length of the spring varies linearly with distance l from the fixed end. Assume also that the mass M of the spring is distributed uniformly along the length of the spring.
1. Calculate the kinetic energy of the spring in terms of M and v.
Hint: Divide the spring into pieces of length dl, find the speed of each piece, find the mass of each piece, and then integrate from 0 to L. The result is not 1/2 M v^2 since not all of the spring moves with the same speed.
2. Now, in a spring gun, a spring of mass 0.360 kg and force constant 4.30 kN/m is compressed 2.00 cm from its unstretched length. When the trigger is pulled, the spring pushes horizontally on a 0.055 kg ball. The work done by friction is negligible. Calculate the ball’s speed when the spring reaches its uncompressed length, ignoring the mass of the spring.
3. Calculate the ball’s speed when the spring reaches its uncompressed length, including the mass of the spring.
4. In part (c), what is the final kinetic energy of the ball and of the spring?