Spring Homework: Calculating Energy Stored

[Smtih]

Homework Statement

The question is:
A spring is hung from a roof. Uncompressed it is 40cm long. The spring is known to have a spring constant of 10Nm-1
A ball of mass 0.2kg is hung from the spring which extends to 60cm. (therefore extension is equal to 20cm)
(Note the ball is stationary not oscillating)
Calculate energy stored in the spring.

Homework Equations

Now to solve this is took the change in gravitational potential energy to find the energy absorbed by the spring. Using E=mgh

A friend used the energy of a spring equation which is E=0.5kx^2

The Attempt at a Solution

Using the change in potential energy I produced the answer of 0.4 J
Using the spring equation an answer of 0.2 J

Which is correct? I'm sure one is incorrect because it does not account for gravity or something along those lines but can't think of a way to show it.

 

[dynamicsolo]

Using the change in potential energy I produced the answer of 0.4 J

And this is correctly calculated
( approx. 0.2 kg · 10 m/(sec^2) · 0.2 m = 0.4 J ). However, it is not the answer to the question. While this is the change in the gravitational potential energy of the ball-Earth system, it does not measure the potential energy stored in the spring.

Using the spring equation an answer of 0.2 J

The change in the spring potential energy
( [1/2] · 10 N/m · [0.2 m]^2 = 0.2 J ) is what the question appears to ask for, so this is probably what should be given for the answer.

The apparent discrepancy seems to be due to the fact that each of you is looking at something different in the overall system. Consider that the ball is subject to two forces: the downward force of gravity and the upward force from the spring. So the work on the ball done by gravity over the downward path is +0.4 J ; the work done by the spring over the same path is -0.2 J . The work done by gravity will be equal to the opposite of the change in gravitational potential energy, which is m · g · delta_y = -0.4 J . There is no net change in the kinetic energy of the ball, so the 0.4 J made available from the gravitational field is taken up in part (0.2 J) by the work done by the spring to keep the ball from accelerating and the rest (0.2 J) goes into the spring's potential energy (at the microscopic level, it goes into stretching the bonds between the atoms in the spring's structure, which is often loosely referred to as "tension").

 

[Moetasim]

I would say that both answers are correct but they are calculating different types of energy. The first answer using the change in potential energy formula is calculating the potential energy stored in the spring due to the gravitational force acting on the ball. This energy is absorbed by the spring and causes it to stretch.

The second answer using the energy of a spring equation is calculating the elastic potential energy stored in the spring due to its spring constant and the displacement of the ball. This energy is what allows the spring to return to its original shape after the ball is removed.

Both of these energies are present in the system and are important to understand when studying the behavior of springs. It is important to note that the energy stored in the spring is not just from the gravitational force acting on the ball, but also from the properties of the spring itself. Therefore, both calculations are correct and provide valuable information about the system.