# 16.49 Springs

Before you read this, I suggest you read posts 16.13, 16.20 and 16.21. Let’s think about a spring that is fixed at one end; when a force pulls it at the other end, it gets longer. When a spring is pulled, so that it gets longer, we say that it is in tension and the force is a tensile force. If the length of the spring was L, when no force was applied, the effect of a force F is to increase the length of the spring to L + ΔL; notice that we are using the symbol Δ to mean “a bit of” or “an increase in” as in posts 16.12, 16.13 and 16.34. So ΔL is the increase in length of the spring. Most springs are designed so that a graph of the applied force, plotted against the extension, is a straight line, as shown above. If the applied force is too great, the spring will become permanently distorted and lose its “springiness”. But in this post, we will consider how the spring behaves when it is not overloaded and behaves as intended.

The slope of the graph of is called the stiffness or the spring constant of the spring. Looking at the graph, you can see that the stiffness is given by k = FL. The higher the value of k, the stiffer the spring. Since we measure force in newtons (N) and distance in metres (m), we measure k in N.m-1 (N/m) (see post 16.13).

When we remove the force, the spring returns to its original length, L. This return to the original length is called recoil.

Why does a spring recoil? When the applied force pulls the spring, it does work (see post 16.20). Since the spring returns to its original length, when the force is removed, this work must be stored as potential energy. The potential energy is then used to return the spring to its original length (see post 16.21), in the same way that an object that has been lifted can use gravitational potential energy to fall (see post 6.21).

How can we calculate the energy in a stretched spring? The force applied to the spring is not constant; it increases from zero to F. The average force to stretch the spring by ΔL is F/2. So the work done is stretching the spring is (F/2) × ΔL = FΔL/2 = (kΔL)(ΔL)/2 = k(ΔL)2/2; the middle step comes from the equation (above) defining k which tells us that F = kΔL. In all this algebra, remember that it is common to leave out the multiplication sign (×) in expressions like kΔL which could also be written as k × ΔL. If all the work done is stored as potential energy, this same result gives us the potential energy of the stretched spring. But using the average value of the applied force, to calculate the work done, only works because the graph of force plotted against extension is a straight line.

The energy stored by a stretched spring can be used to do work; for example, springs were once used instead of batteries as the source of energy for clocks and watches. We still call a motor which uses a stretched spring a clockwork motor. However, a clockwork motor uses a flat spiral spring that is tightly wound; as the spring unwinds, it releases the energy used to wind it. A spring gets shorter when a force pushes down on it. When the spring gets shorter, we say that it is in compression and that the applied force is a compressive force. We call the change in dimensions of a spring in either tension or compression a deformation. If we are going to use a spring in compression, we must make sure that it doesn’t bend out of shape when it is compressed (see picture above). This bending, which arises from unequal compression, is called buckling.

We can easily distinguish between the directions of tensile and compressive forces by giving one a positive value (conventionally for tension) and the other a negative value (for compression). Then ΔL is positive for a stretched spring and negative for a compressed spring.

Understanding compression and tension is also essential for understand the behaviour of structures like beams and suspension bridges. And it shows the importance of the direction of a force because the direction determines whether the spring gets longer or shorter.

Related posts

Follow-up posts

20.2 Deformation of objects