Hooke's Law

Hooke's Law deals with what happens when you stretch or compress elastic objects like springs. It's one of those really useful bits of physics that turns up everywhere - from the suspension in cars to trampolines to weighing scales. Understanding Hooke's Law means understanding how objects respond to forces and how energy gets stored when things deform.

Forces and Changing Shape

If you want to stretch, bend, or compress something, you can't do it with just one force - you need at least two forces working together.

Think about stretching a spring. If you just pull on one end, the spring will move with you - it won't stretch. You need to hold one end fixed (that's one force) while you pull on the other end (that's the second force). These two forces working in opposite directions are what actually cause the spring to extend.

The same goes for compressing something or bending it - you always need more than one force.

Elastic Deformation vs Inelastic Deformation

When you apply forces to stretch an object, two things can happen:

• Elastic Deformation - This is when the object returns to its original length when you remove the force. Think of stretching a rubber band and then letting go - it springs back to its original size.

• Inelastic Deformation (also called plastic deformation) - This is when the object does NOT return to its original length after you remove the force. It stays permanently stretched or deformed.

Hooke's Law - The Main Principle

The extension of an elastic object (like a spring) is directly proportional to the force applied(), provided that the limit of proportionality is not exceeded - this means that the object will return to it's original length when the force is removed.

• "Directly proportional" means that if you double the force, you double the extension. If you triple the force, you triple the extension. It's a linear relationship - if you plotted it on a graph, you'd get a straight line through the origin.

• Go beyond the "limit of proportionality", and the relationship breaks down. The spring might still stretch, but it won't follow Hooke's Law anymore.

Formula:

Note that this equation depends on the extension (), not length. Extension is how much longer the spring has become - so if a spring was originally long and you stretch it to the extension is (or when converted to metres).

What is the Spring Constant?

The spring constant () tells you how stiff a spring is. A higher spring constant means a stiffer spring - you need more force to stretch it by the same amount.

• A spring with is much stiffer than a spring with .

• To stretch the stiff spring by , you'd need to apply of force, but only for the less stiff spring.

• Springs have different spring constants depending on their uses.

  • A spring in a car's suspension system has a very high spring constant (it's very stiff) because it needs to support the weight of the car.

  • The spring in a retractable pen has a much lower spring constant.

Hooke's Law for Compression

Everything we've said about stretching also applies to compression. If you compress a spring (squash it down), the relationship still holds, but now '' represents the compression rather than the extension.

Example: My dog is pulling on her bungee lead. The bungee extends from to and has a spring constant of . I can calculate the force my dog is pulling with using Hooke's Law, provided that the bungee lead is being elastically deformed.

First, I need to find the extension of the lead:

Then I can use Hooke's Law:

When you plot force (on the y-axis) against extension (on the x-axis) for a spring, you get a force-extension.

• At the start, you get a straight line through the origin. This straight-line section is where Hooke's Law applies - where force and extension are directly proportional. This is called the linear region.

• The gradient (slope) of this straight line is the spring constant, . A steeper line means a bigger spring constant (stiffer spring).

The Limit of Proportionality

If you keep increasing the force, eventually you reach the limit of proportionality. This is the point beyond which the spring stops obeying Hooke's Law. On the graph above, this is where the line stops being straight and starts to curve.

Beyond the limit of proportionality, you need more and more force to produce the same increase in extension - the spring is getting harder to stretch. Eventually, you reach the elastic limit - the point where the spring becomes permanently deformed and won't return to its original length.

Linear vs Non-Linear Relationships

A linear relationship between force and extension means the graph is a straight line - Hooke's Law is obeyed. Force and extension are directly proportional.

A non-linear relationship means the graph is curved - Hooke's Law is not obeyed. This happens when you go beyond the limit of proportionality.

The practical to investigate Hooke's Law includes the following steps:

• Measure the original length of the spring using a ruler

• Attach known mass to the spring

• Measure the new length of the spring using a rule

• Calculate extension by subtracting the original length from the new length

• Calculate force by using where is the mass of the mass attached and is the gravitational field strength ()

• Repeat steps by adding more masses

• Plot a graph of extension against force

• If the graph is linear and passing through the origin, the spring obeys Hooke’s Law

insert Hooke's Law set up

To improve this experiment, you could:

• clamp the ruler

• ensure you read the ruler from eye level

• ensure you wear safety goggles in case the spring breaks