Hooke's Law

Hooke's law describes how elastic objects, such as springs, behave when stretched or compressed.

If you want to stretch, bend, or compress something, you need at least two forces working together. Think about stretching a spring. If you just pull on one end of a spring at rest on the ground, the spring will just move with you - it won't stretch. You need to hold one end fixed while you pull on the other end. These two forces, acting in opposite directions, would cause the spring to extend.

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.

The extension of an elastic object (like a spring) is directly proportional to the force applied, up until the limit of proportionality.

• "Directly proportional" means that if you double the force, you double the extension. If you triple the force, you triple the extension. If you plotted a graph, a straight line of best fit through the origin would show you that the variables are directly proportional.

• The "limit of proportionality" is the point where Hooke's law is no longer obeyed; the spring will still stretch, but the amount it extends is not proportional to the force applied.

Formula:

-> Extension (), is not the same as length. Extension is how much longer or shorter the spring has become. It is the change in length when the force is applied.

-> Spring constant () tells you how stiff a spring is. A higher spring constant means a stiffer spring - you will need a greater force to stretch it by the same amount. 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 must support the car's weight.

• The spring in a clicky pen has a much lower spring constant, as only a little force applied by your thumb is needed to compress it.

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:

To investigate Hooke's Law, to show that force is directly proportional to extension, , set up the equipment as shown below and follow the method:

• Measure the original length of the spring using a ruler

• Attach a known mass to the spring

• Measure the new length of the spring using a ruler (ensure to read from the bottom of the spring, from the pointer, at eye level, to avoid parallax error)

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

• Calculate the force applied to the spring using the equation, , where is the mass of the mass attached and is the gravitational field strength ()

• Repeat all steps above by adding more masses to increase the force applied

• Plot a graph of the results, plotting force against extension

Results

When the results are plotted, and a line of best fit is drawn through the data, we get a graph like the one below:

• Initially, there is a straight line through the origin. This straight-line section is where Hooke's Law is obeyed -> force and extension are directly proportional.

• The gradient (slope) of this line gives us the spring constant, , as . A steeper line, therefore, means the spring has a bigger spring constant (stiffer spring).

• We can see that the line eventually curves as more and more force is applied to the spring. The point at which the line starts to curve is known as the limit of proportionality. This is the point at which the spring stops obeying Hooke's Law. The spring is no longer elastic, and is now known as plastic.