Question

My query is that suppose an ideal spring is present and is compressed from both the sides by a distance x1

from the left and a distance x2
from the right. So what will be the potential energy stored in the spring at this moment?

Using the formula 12kx2
, should it be 12k(x1+x2)2
or the sum of individual energies 12k(x1)2+12k(x2)2
? According what I have read it should be the individual sum.

But I can't understand why as if we stretch a spring, say to y1
and y2
from both side, the force applied by spring on both sides will be −k(y1+y2)
, as the spring is ideal, and there can't be net force on it, so it will apply same force on both side. But why should the potential energy be measured individually rather taking the sum of total compressed/stretched distance from both sides? Options:
A)
12
�
(
�
1
+
�
2
)
2
12k(x
1
​
+x
2
​
)
2

B)
12
�
(
�
1
)
2
+
12
�
(
�
2
)
2
12k(x
1
​
)
2
+12k(x
2
​
)
2

C)
12
�
(
�
1
�
2
)
12k(x
1
​
x
2
​
)
D)
12
�
(
�
1
−
�
2
)
2
12k(x
1
​
−x
2
​
)
2

Answer 1

B) 12k(x₁)² + 12k(x₂)²the potential energy stored in an ideal spring compressed from both sides should be calculated by adding the individual potential energies of the respective compressions, leading to 12k(x₁)² + 12k(x₂)².

Explanation:

The potential energy stored in an ideal spring compressed from both sides by distances x₁ and x₂ should be the sum of the individual potential energies, given by the formula 12k(x₁)² + 12k(x₂)². This is due to the nature of potential energy in springs, which is directly proportional to the square of the displacement from the equilibrium position. Each side of the spring compresses independently, resulting in distinct potential energies on each side, hence the sum of their individual energies represents the total potential energy stored in the spring system.

When considering the compression or extension of an ideal spring from both ends, the force exerted by the spring on each side is equal in magnitude but opposite in direction. However, when calculating potential energy, it's crucial to account for each displacement separately because the potential energy of the spring is a function of the square of the displacement. Therefore, the sum of the squares of x₁ and x₂ provides an accurate representation of the energy stored in the spring due to compression from both ends.

In conclusion, the potential energy stored in an ideal spring compressed from both sides should be calculated by adding the individual potential energies of the respective compressions, leading to 12k(x₁)² + 12k(x₂)².