Block energy question

Question

A block is compressed a distance x against a spring such that the system now has a total mechanical
energy of 100J. Assume there are no frictional forces.

 

A) How many joules of energy is in the form of potential energy at this position? How much energy
is in the form of kinetic energy?

 

B). The block is released, and the spring has expanded to
half the original compression (the new position is at x’=x/2). How much energy is in the form of potential
and kinetic energy? Note we are not asking for the velocity

 

C). What is the Potential Energy of the spring when x’’=0, the un-stretched position? What is the Kinetic
Energy of the block at x’’=0?

Answer 1

A) Since the block is still being pressed into the spring, it is not moving, so there is no kinetic energy. Therefore all of the energy stored in the system is potential.

 

B) Elastic potential energy is governed by PE_s = 1/2 kx² where k is the spring constant of the elastic object, and x is the distance it is deformed from its rest position. If you plug in x/2, you get:

PE_s = 1/2 k (x/2)² = 1/2 k x²/4 = 1/4 (1/2 k x²)

The potential energy is 1/4 of what it started at, 1/4*100 = 25J. Energy is conserved, so the rest of the energy must now be kinetic: 75J

 

C) When x'' = 0, PE_s = 1/2 k (0)² = 0. All of the energy is now kinetic.