# Elastic Strings and Springs

Potential energy comes in many forms. Sometimes it's stored in chemical bonds, sometimes it's stored in an object being acted on by a planetary body's gravitational field, and sometimes, it's stored in something as simple as a coiled piece of metal. How is that? Well, it's due to something called elasticity.

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## Definition of elasticity

Elasticity is a special physical property that can be found all over the place. Elastic bands are elastic (obviously), springs are elastic, the shaft of a bow is elastic, the shaft of a golf club... the list goes on. But what exactly is it that all these things have in common; what is elasticity?

Elasticity is the ability of an object to resist distorting forces and to return to its original shape.

Take the example of a plastic ruler, in fact, if you have one in front of you - even better! It's easy to bend the ruler a little bit, but the more you bend it, the harder it becomes to bend it further. This is the ruler's resistance to distortion. Now, if you were to let the ruler go, it snaps back straight - this is it returning to its original shape.

Now, let's think about a cracker from your cupboard. It gives a small amount of resistance to being distorted, but then just snaps in two - it does not return to its original shape, and is not very elastic at all.

Simple enough right? Feel free to try doing this with other objects you can find around the house (being careful not to break anything) to see which have elastic properties and which don't!

Hopefully, you now have a good idea of what elasticity is, but how exactly does it relate to potential energy?

## 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.

For more information, see the article Kinetic and Potential Energy.

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.

More details are in found in Work Done by a 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.

Tension is the force transmitted through an object such as a rope or string that acts against forces pulling it apart.

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.

An object is hung from a spring scale with a spring constant of $$80\, \text{N}\,\text{m}^{-1}$$. The spring is distorted by $$6\,\text{cm}$$. What is the mass of the object, assuming the mass of the spring is negligible?

Solution:

First, consider Hooke's Law, and substitute in the known values to find the force of the spring at equilibrium. Remember, displacement must be in terms of metres, not centimetres. Then substituting into the equation gives you

\begin{align} F &= -kx \\ &= -80(-0.06) \\ &= 4.8 \, \text{N}. \end{align}

The displacement of the spring is negative since it's downward.

Since the force of the spring is $$4.8\,\text{N}$$ when the spring and object are in equilibrium, the weight of the object must be of equal magnitude (and opposite direction), or in terms of the variables:

$W = -4.8 \, \text{N}.$

Now, assuming the measurement was taken near Earth's surface, the acceleration due to gravity will be $$-9.81 \, \text{m}\,\text{s}^{-2}$$, so the mass can be calculated using Newton's second law:

\begin{align} F &= ma \\ W &= mg \\ -4.8 &= m (-9.81), \end{align}

giving you

$m = 0.49 \, \text{kg} ,$

where the answer has been rounded to two decimal places.

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.

A mass of $$5\, \text{kg}$$ attached from above to a tension spring with spring constant $$40\,\text{N}\,\text{m}^{-1}$$ is dropped vertically downward. It then oscillates before reaching equilibrium. What is the period of oscillation?

Solution:

Use the formula for the period of oscillation of a spring:

$T = 2\pi\sqrt{\frac{m}{k}}.$

Substitute the known values for $$m$$ and $$k$$ in to find the time period, $$T$$, giving you

\begin{align} T &= 2\pi \sqrt{\frac{5}{40}} \\ &=2.22 \, \text{s} .\end{align}

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.

A car is attached to an elastic cable, attached to a wall. The owner of the car wants to measure the maximum horizontal force that the car is capable of generating, so has it pull against the elastic cable, and measures its displacement.

The elastic cable has a spring constant of $$300\, \text{N}\,\text{m}^{-1}$$, and an unstretched length of $$5\,\text{m}$$. The car manages to drive $$16\,\text{m}$$ from the wall. How much horizontal force can the car exert?

Solution:

This problem can be solved using Hooke's Law, $$F = -kx$$. The displacement when the car and cable are in equilibrium, $$x$$, of the cable is the difference between the final distance of the car from the wall, and its unstretched length, so

$x = 16-5 = 11\, \text{m}.$

Now, use Hooke's law to calculate the force exerted by the cable when the car and cable are in equilibrium:

\begin{align} F &= -300(11) \\ &= -3300\, \text{N}. \end{align}

The force exerted by the car is equal and opposite to that of the rope at equilibrium, therefore the car can exert $$3300\, \text{N}$$.

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

A person of mass $$60\,\text{kg}$$ attached to a bungee rope jumps off a cliff. The rope has a spring constant of $$50\,\text{N}\,\text{m}^{-1}$$, and unstretched length of $$15\,\text{m}$$.

After the person jumps, they oscillate before reaching the equilibrium position. During oscillation, the rope never goes slack. Assume the mass of the rope is negligible.

a) What was the period of oscillation of the person?

b) What was the length of the rope at equilibrium?

Solution:

a) The period can be calculated using the equation:

\begin{align} T &= 2\pi \sqrt{\frac{m}{k}} \\ &= 2\pi \sqrt{\frac{60}{50}} \\ &= 6.28\,\text{s} .\end{align}

b) Using Hooke's Law, $$F=-kx$$, the displacement of the rope from its unstretched position can be found. The force of the rope at the equilibrium position is equal and opposite to the weight of the person, giving you

\begin{align} W &= mg \\ &= 60(-9.81) \\ &=-588.6\,\text{N}, \\ F&= -W \\ &= 588.6 \,\text{N}. \end{align}

The force can be substituted into the Hooke's Law formula along with the spring constant to find $$x$$:

\begin{align} 588.6 &= -50x \\ x &= -11.77\,\text{m} .\end{align}

The length of the rope at the equilibrium position is the sum of the magnitude of this displacement and the unstretched length of the rope:

\begin{align} L &= 15 + 11.77 \\ &= 26.77\,\text{m}. \end{align}

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?

## Elastic strings and springs - Key takeaways

• Springs and elastic strings are both elastic objects.
• Elasticity is the ability of an object to resist distorting forces and to return to its original shape.
• Hooke's Law states that the force exerted by an elastic string or spring is proportional to its distortion: $$F = -kx.$$
• The resting position of an object hanging from an elastic string or spring is dependent on the weight of the object and the stiffness of the string or spring.
• The period of oscillation of an elastic string or spring attached to a mass can be found with the following formula: $T = 2\pi \sqrt{\frac{m}{k}}.$

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##### Frequently Asked Questions about Elastic Strings and Springs

What is the difference between a spring and a string?

A spring is generally a coil of metal that displays either compression or tensile elastic qualities - an elastic string is a string or rope that exhibits tensile elastic properties.

Does a string have elastic energy?

An elastic string can store elastic energy when work has been done in stretching it.

Why is the string elastic?

A string is elastic if it returns back to its original size and shape after being stretched and let go.

When elastic strings are stretched beyond their natural length the tension is measured by which law?

Hooke's Law.

Where does the spring have the most elastic potential energy?

A spring stores most of its elastic potential energy wherever it is most compressed (for a compression spring) or stretched (for a tension spring).

## Test your knowledge with multiple choice flashcards

What is the equation for the time period of an oscillating elastic string or spring?

What is the Hooke's Law formula?

What is the resting position of an object hanging from an elastic string or spring is dependent on?

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