Is the work done by a spring and the work done on a spring the same?

Lets imagine we have a spring of a linear characteristic such that the force the spring exerts is proportional to its deformation $$\delta$$. The spring is attached horizontally to a wall, now if we stretch the spring by applying a horizontal force $$\vec{F}$$ the work done on the spring will be: $$W_{1\rightarrow 2}=\frac{k}{2}(\delta_{2}^{2}-\delta_{1}^{2})$$ where $$\delta_2$$ and $$\delta_1$$ are the final and initial deformations of the spring respectively and $$k$$ is the spring proportionality constant.

Now suppose we let go of the spring, the work done by the spring will be: $$W_{2\rightarrow 1}=-\frac{k}{2}(\delta_{2}^{2}-\delta_{1}^{2})$$

My question is why is the work now negative?

The work done on an object and the work done by an object are clearly not the same thing: in the former case it is the work done by the force acting on the object, whereas it is the work done by the force produced by the object itself. It is often the case that these two forces are related via newton's second law or energy conservation (like in the question) and therefore have equal magnitude and opposite direction - then they do work of equal magnitude, but having different sign.

Btw, there may be a problem with notation in the question: shouldn't the second equation be for $$W_{2 \rightarrow 1}$$ instead of $$W_{1 \rightarrow 2}$$, if $$1$$ and $$2$$ refer to the states where the string unstretched and stretched respectively?

• yes, you're right ill change the indexes. Sep 1, 2020 at 14:43

The change in sign indicates that the sense in which energy flows is reversed.

First, the work applied to stretch the spring increases the internal energy of the spring by adding potential energy to it. Later, when the force that holds the spring in a stretched configuration is removed, the energy is released and transferred elsewhere (e.g., by being dissipated as thermal energy in your muscles). In the end the potential energy of the spring is not there anymore because it has been transferred through mechanical work to the environment.

This can be further confirmed if one imagines that the processes of stretching/unstretching are done very slowly and such that one looks like the time-reversed version of the other. The forces applied will be the same, but note that in the stretching case both the force and the displacement rate point in the same direction (positive work on the sping = energy gain by the spring), whereas in the unstretching phase they point in opposite directions, producing negative work (energy loss by the spring).

If you can ignore friction then yes. A spring force is a conservative, or more specifically the potential energy stored in the spring is conservative.

However much you put in, you can extract out.

This is in contrast to a heat engine whose work extracted is always less than the energy put in.

The work done by the spring is positive when we let go of the spring and let it return to its initial state. The work done by the spring as shown by the second equation is a positive quantity, despite the negative sign before the k. This is because the final deformation of the spring is less than the initial deformation, the quantity in the parenthesis is negative and the negative signs cancel out to give a positive quantity.

As an aside, the work done by a spring and the work done on a spring are equal in magnitude and opposite in sign. So when a spring is streched by an external force and positive work is done on the spring, the work done by the spring (due to the reaction force in the spring) is negative. In the case considered above where the work done by the spring is positive, the work done on the spring is negative.

I have come to the conclusion that before deciding to derive any type of energy related equation its important to choose a system for the derivation. Using a sign convention to indicate positive and negative work on a system is key, I will use the convention used in thermodynamics where work done on a system is negative and work done by the system is positive. By combining these two concepts we can conclude that if a force is applied to the spring the work on the spring will be negative and if the spring applies a force on its surrounding the work will be positive (of course assuming that the system is made up of just the spring).

Both are the two different things workdone on the spring means that the work is done by an external agent in compression or elongation of the spring by applying some force against the spring force which can be variable or constant and it depends upon your choice. Whereas workdone by the spring is the different thing, just consider the case of SHM with spring block system when the block is at the extreme position and turns back automatically towards the mean position of the block (considering at $$x = 0$$) spring is doing the work that is variable with the distance, you will get:-

$$W = \int{\frac{1}{2}kx^2dx}$$

Here, $$k$$ is the spring constant.