Elastic Strings and Springs

Elastic potential energy and work done on elastic strings and springs

Let's consider an elastic band, which is one kind of elastic string. Once again, if you have one in front of you, even better! If you take an elastic band and gently stretch it out, then let go, what happens? Well of course it pings back together. Everyone has seen this when they played with elastic bands as a kid (or as an adult). The elastic band moved when it was let go, in fact, it moved quite fast, so where did the kinetic energy come from?

When you lift a rock off the ground, you apply a force to it to raise it and it gains gravitational potential energy. Elastic objects are just the same. When you stretch them, you are applying a force to them, and as they are stretched they store elastic potential energy. When the elastic object is let go, the elastic potential energy converts to kinetic energy resulting in movement.

Remember that when an object is acted on by a force, \(F\), over a distance, \(d\), work, \(W\), is done on that object. Work is just a special name for the energy that is transferred to an object by the application of force.

So the energy transfer when you stretch and let go of an elastic band looks like this:

\[\text{Your Energy }\xrightarrow{\text{Work done}} \text{Elastic Potential Energy} \xrightarrow{\text{Band let go}} \text{Kinetic Energy}\]

You deposited a little bit of your own energy into the elastic band so that it would ping when you let go. Pretty cool, right?

You might remember that the equation for work is

\[W = Fd,\]

where \(W\) is in joules, \(F\) is in newtons, and \(d\) is in metres. How much energy the elastic band obtains then is down to two things: how much it is stretched (the distance); and the total amount of force applied to it.

An important thing to remember about elastic objects is that the more elastic potential energy they gain, the more they want to return to their original position. You can feel this in the elastic band, the further it is stretched, the harder it becomes to stretch it even more. It's just the same with springs. Why is this? Well, it's all to do with a special force called tension.

Tension in elastic strings and springs

What is tension then? No, it's not just the feeling you get when you're having a really awkward conversation. It's actually an important property in physics.

Consider a regular, non-elastic rope. When you pull it apart with your hands, it goes taut, it almost feels as if the rope is pulling against you. Well, that's because it is. Newton's third law tells us that every action has an equal and opposite reaction. Tension is the reaction of the rope to you trying to pull it apart, it acts against you, towards the centre of the rope.

Fig. 1 - Tension in a rope

Eventually, if the tension in the rope becomes great enough, the rope will snap. If you've ever pulled a Christmas cracker, you've seen this exact concept in action.

Ok, so how does all of this relate to elastic objects like elastic strings and springs? Well, elastic objects are special, because, unlike the rope, once they are taut, they can be pulled apart even further. The further apart they are pulled (or sometimes with a spring, the more it is compressed), the greater the force of tension becomes.

That, in a nutshell, is why the more you stretch an elastic object, the harder it becomes to stretch it further. So, what actually is the deal with elastic strings and springs, specifically? Let's take a closer look.

Relationship and differences between elastic strings and springs

The concepts covered so far to do with elasticity can be applied to both elastic strings and springs, but there are differences between the two that are important to keep in mind when tackling problems.

Elastic strings

An elastic string is really just what it sounds like. Like a regular string, an elastic string is fully flaccid until pulled taut, but unlike a regular string, an elastic string can return to its original shape after being distorted and can store elastic potential energy.

An elastic band is essentially just an elastic spring in a loop. The important thing to remember is that an elastic string can only store elastic energy by being stretched, not compressed.

You may be asking then what sort of an object can store elastic potential energy by being compressed. Well, as it happens you're about to find out.

Springs

Springs, similar to elastic strings, are objects capable of storing elastic potential energy, however, they differ in how that energy is stored. Springs are typically helical coils of metal, and can be constructed such that they store elastic potential energy when stretched (tension springs) or when they are compressed (compression springs).

Compression springs, as you can imagine, are generally used to keep objects apart, and tension springs are generally used to keep objects together. The forces in spring can be a bit complex to consider, but the important thing is, much as with an elastic string, the axial (along the central axis) force the spring exerts acts to return it to its original shape. This is referred to as the restorative force.

Fig 2. - Forces in tension and compression springs

Even better, there's a nifty little relationship called Hooke's Law that can be used to find what this force will be:

\[F = -kx,\]

where \(F\) is the total restorative force exerted by the spring in newtons, \(k\) is the spring constant of the spring in newtons-per-metre, and \(x\) is the displacement of the spring from its original shape in metres.

The spring constant of a spring can basically be thought of as a measure of that particular spring's stiffness - its resistance to distortion. This concept of spring stiffness can also be applied to problems with elastic strings.

Hooke's Law is a super useful relationship. Luckily we have an entire explanation dedicated to it - so head over and take a look if you're interested in learning more!

Vertical elastic strings and springs

So, now that the basics of elastic strings and springs have been covered, let's take a look at a slightly more specific scenario.

It's the eighteenth century, and you want to weigh some stuff. You look around at the options available for weighing devices and think, hmmm, I can do better. How could you put your knowledge of springs to use?

Well, Richard Salter thought why not hang stuff from the spring? He hung stuff from a tension spring, and as expected the heavier stuff sank lower than the lighter stuff. This is a natural extension of Hooke's Law, the weight of the object acts downwards and does not change, and the tension of the spring acts upward, and gets larger the lower the object sinks. Once the force of tension in the spring becomes equal to the weight of the object, the system is in equilibrium and the object and spring hang still.

All he had to do was measure how far the spring distorted, and hey presto, he could calculate the mass of the object. Let's take a look at an example to see how he did this.

This is how you can consider elastic strings and springs vertically; by taking into account the effects of gravity. But there's a little more to the story. This scenario only takes into account the position of the spring in equilibrium, but what does the motion of an elastic spring or string look like as it approaches that point? Well in some scenarios, it will just stretch out until it reaches that point and stops - but in others, it will bounce up and down before settling. We call this oscillation. Let's take a closer look.

Time period of elastic strings and springs

Without getting too technical, a spring or elastic string will oscillate when the speed of its stretch (or compression, in compression springs) is too great. This is because the increasing force of the spring or string is unable to slow the stretch down to a stop by the time it reaches the equilibrium position. The spring will then bounce up and down around the equilibrium point until it eventually stops there.

When this phenomenon occurs, the spring and mass system is referred to as underdamped.

A property that can be important to find in engineering applications is the time period of a spring system's oscillation. Luckily, there's a handy little equation that can be used to find just that:

\[T = 2\pi\sqrt{\frac{m}{k}},\]

where \(m\) is the mass on the end of the spring in kilograms, \(k\) is the spring constant in newtons-per-metre, and \(T\) is the time period of the oscillation in seconds.

Perhaps counterintuitively, gravity does not affect the time period of oscillation, only where the equilibrium position is located - this is because it is a constant downward force, not changing with the position of the spring.

Let's take a look at an example for a bit of practice.

Not so hard right? Let's go on and take a look at some more examples to make sure you've got everything.

Elastic strings and springs examples

Let's start off with a question on Hooke's Law.

Did you know that bungee jumping also works on the principles of elasticity? Let's take a look at another example to see how.

There you have it, a little introduction to elastic strings and springs! If you want to really get stuck into the details of elasticity and Hooke's law, why not check out our other explanations?