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]
