Four springs are stretched to the same distance from the equilibrium position. The spring constants are listed in the table.

\begin{tabular}{|c|c|}
\hline Spring & Spring Constant (N/m) \\
\hline W & 24 \\
\hline X & 35 \\
\hline Y & 22 \\
\hline Z & 15 \\
\hline
\end{tabular}

Which lists the springs based on the amount of elastic potential energy, from greatest to least?

A. X, Y, W, Z
B. X, W, Y, Z
C. Z, W, Y, X
D. Z, Y, W, X



Answer :

To determine which list of springs correctly represents the order from greatest to least elastic potential energy, we need to keep in mind that the elastic potential energy [tex]\( U \)[/tex] stored in a spring is given by the formula:

[tex]\[ U = \frac{1}{2} k x^2 \][/tex]

where [tex]\( k \)[/tex] is the spring constant and [tex]\( x \)[/tex] is the displacement from the equilibrium position. Given that all the springs are stretched to the same distance [tex]\( x \)[/tex], the only variable that affects the order of the elastic potential energy is the spring constant [tex]\( k \)[/tex].

Here are the given spring constants:
- [tex]\( W : 24 \, \text{N/m} \)[/tex]
- [tex]\( X : 35 \, \text{N/m} \)[/tex]
- [tex]\( Y : 22 \, \text{N/m} \)[/tex]
- [tex]\( Z : 15 \, \text{N/m} \)[/tex]

Since the elastic potential energy is directly proportional to the spring constant (assuming the same displacement [tex]\( x \)[/tex]):

1. First, rank the springs in order of their spring constants from greatest to least:
- [tex]\( X \)[/tex] has the highest [tex]\( k \)[/tex] value of 35 N/m.
- [tex]\( W \)[/tex] is next with a [tex]\( k \)[/tex] value of 24 N/m.
- [tex]\( Y \)[/tex] follows with a [tex]\( k \)[/tex] value of 22 N/m.
- [tex]\( Z \)[/tex] has the lowest [tex]\( k \)[/tex] value of 15 N/m.

2. With this ranking, the order of the springs from greatest to least in terms of elastic potential energy will be:
- First: [tex]\( X \)[/tex]
- Second: [tex]\( W \)[/tex]
- Third: [tex]\( Y \)[/tex]
- Fourth: [tex]\( Z \)[/tex]

Thus, the correct order of springs based on the amount of elastic potential energy stored, from greatest to least, is:

[tex]\[ X, W, Y, Z \][/tex]

The correct answer is:
[tex]\[ X, W, Y, Z \][/tex]