# Why the work done by system not stored as potential energy?

Now choose a spring mass system now work done by external agent in slowly moving from equilibrium position is stored as potential energy but where is work done by spring force gone.For genralization work done on system is stored as potential energy but where is work done by system gone during the process?

For generalization work done on system is stored as potential energy but where is work done by system gone during the process?

The work done by the spring mass system takes the work done on the spring mass system and stores it as potential energy in the spring mass system. The net work done on the mass of the system is zero. The following is presented as an explanation.

When an external agent (for example you) applies a force to a mass on the end of a spring (let's assume the spring is horizontal so that gravity plays no role) it compresses the mass spring system. The force you apply to the mass is in the same direction as the displacement of the mass so you do positive work on the mass equal to $$\frac{kx^2}{2}$$.

At the same time the spring exerts a force on the massin a direction opposite the displacement of the mass. Therefore it does negative work taking the energy you provided to the mass and storing it as potential energy of the spring mass system. The net work done on the mass is zero, since by the work energy theorem the net work done on an object equals its change in kinetic energy. For this example, the change in kinetic energy is zero since the mass begins and ends with zero velocity.

Bottom line, the work you did winds up stored in the spring mass system as potential energy. The work you did in compressing the spring is not gone. It just becomes potential energy.

The gravity analogy is when you lift an object starting at rest on the ground and bring it to a point at rest a height $$h$$ from the ground you do positive work transferring energy to the object. At the same time the force of gravity, which acts opposite the direction of movement of the object does an equal amount of negative work, taking the energy you gave the object and storing it as gravitational potential energy of the object/earth system.

Hope this helps.

• Hello @BobD, I have a related question. If there wasn't any block attached to the spring and the force was directly applied by me on the spring, then the force applied by me does a +ve work trying to increase the KE of the spring but then there is no increase in the KE of the spring, which force is responsible for doing a -ve work in this case, which tries to remove the energy supplied by the +ve work done by me? Commented May 23, 2022 at 14:12
• @HarshitRajput In this, like most spring problems, the spring is considered to be massless to simplify the analysis. A massless spring cannot possess KE. Once one includes the mass of the spring, the analysis becomes more complex, as you can see in the following link. physics.stackexchange.com/questions/471462/… Commented May 23, 2022 at 15:28

Well consider the definition of potential energy here:-

The change in potential energy of a system is defined as the negative of work done by the internal conservative forces of the system.

In your particular case it is the spring force which is internal and is responsible for change in potential energy of the spring.

The work done by external agent is equal to change in potential energy of the system if and only if the change in kinetic energy of the system is zero.

Again, I want to emphasize that when spring is pushed or extended slowly only then the work done by external agent is equal to change in potential energy of the spring system.

• Yes again. Yet another example of why definitions of potential energy in terms of the external force are at best confusing, and possibly faulty. Unfortunately, many introductory texts use the faulty definition. Commented May 28, 2019 at 20:25
• Why is it necessary that the process be quasi-static (assuming an ideal spring)? Isn't it only necessary that the kinetic energy at the start of the displacement is the same as at the end? Commented May 28, 2019 at 20:38

No energy is lost.
An external force does positive work on the spring when the spring is compressed because the external force on the spring and the displacement of that force are in the same direction.
You can also say that the spring does an equal amount of negative work when the spring is compressed because the force exerted by the spring and the displacement of that force are in opposite directions.
The work done on the spring is equal to the work done by the spring and so there is no net work done on the spring.
And yet another way of explaining what has happened is to say the the spring has gained elastic potential energy as it was compressed equal to the positive work done by the external force (or equal to the negative work done by the spring force).

To add to the previous answer. The energy stored is known as Elastic Potential Energy to the best of my knowledge. Thus, the energy is in fact stored as a form of potential energy - since the position of the mass in the mass-spring system determines the kinds of energy it possesses.

After the spring is extended by x (until this, work is done on the spring, i.e. not by the spring), this spring then returns back to its natural length for which it does some work that is $$\frac{1}{2}kx^2$$. This also applies for the case when the spring is compressed.