A spring is cut into 4 pieces and energy stored by compression. What is the ratio of the displacement required to store the same energy in the original spring and the pieces.
The potential energy stored in a spring that is either compressed or pulled to change its length from the equilibrium length by x is given as PE=(1/2)*k*x^2 where k is the spring constant.
When a spring is cut the spring constant of each of the pieces is the same as the spring constant of the original spring.
Here, the original spring is being cut into 4 equal pieces and all are being compressed by a length x to store energy in them. It is assumed they are not in series as that would make it the same as the original spring. When the 4 pieces are placed in parallel, the equivalent spring constant of the entire system made of the springs changes and we get an equivalent spring constant of 4k.
When the original spring is compressed by x, the energy stored is (1/2)*k*x^2. If the compression required to save the same energy in the system made by the 4 springs is x', we have PE = (1/2)*(4k)*x'^2
Equating the potential energy stored gives
(1/2)*k*x^2 = (1/2)*(4k)*x'^2
=> (x/x')^2 = 4k/k
=> x/x' = 2
The ratio of the displacement of the original spring and the displacement of the system of the 4 pieces to store the same amount of energy is 2:1.