Question

In each of the cases shown in the figure, the block has been displaced by the same amount from equilibrium. Rank the cases based on the elastic potential energy stored in the springs.

Answer 1

@ganeshie8

Answer 2

Hi! I'm going to refer to these connections like they do in electronic circuits.
If I say that the springs are "in series," I mean that they are one after another (like a book series).
If I say springs are "in parallel," I mean that they are connected so that are off in their own branches. In the diagrams, they are even parallel to one another.
The idea is to find a spring that replaces a chunk of springs but leaves the same effect. You have to replace parallel springs first, then series.

I base this on four things:
1. Displacement is the same, so the springs in series have displacements that add up to the same.
2. The springs that are in parallel are stretched the same length.
1. For series, the force is the same throughout all springs, because of Newton's third law.
2. For parallel, each spring has its own force, dependent on the spring constant and displacement. However, look to a simple force diagram to understand that they just add up.

The parallel ones are easy, they have the same displacement, and each \(F_i=k_ix\), so the force is a total of
\(k_1x+k_2x+...=(k_1+k_2+...)x\).
This means our new

Series ones are tougher, because their displacements depend on their spring constants.
But think: the same force is imparted, and the total displacement is the sum of each individual. So,
\(F_1+F_2+F_3=F_{total}=3F\)
\(k_1x_1=k_2x_2=k_3x_3\)Solve for \(x_1\)

Since I need to sleep, you can actually look at this: http://en.wikipedia.org/wiki/Series_and_parallel_springs

It looks good! Just know that when you take two springs in series, you can then view it as one spring with a certain \(k\). Then you can do the same thing to the next.
So, the \(\dfrac1{k_eq}=\dfrac1{k_1}+\dfrac1{k_2}\) can easily be extended for three springs in series, \(\dfrac1{k_eq}=\dfrac1{k_1}+\dfrac1{k_2}+\dfrac1{k_3}\). You don't have three springs in series, though, so you should be fine.

Take care, and good luck.