Hooke's Law states that the force needed to extend a spring by a certain distance is directly proportional to the distance extended. This is shown like this:

F = k × e

Where:

F = force applied (in N)

k = spring constant (stiffness of the spring in N/m)

e = distance extended by spring (in m)

A spring is essentially an elastic object able to store mechanical energy.

Hooke's law is an explanation of elasticity – which is the tendency of an object to restore itself to its original shape after distortion. This force created by the object in returning to a normal shape after experiencing distortion can be referred to as a "restoring force". When extending a spring, this is the force that must be overcome- and Hook'es Law states that it is directly proportional to the length you extend the spring by.

Say, for example, we are conducting an experiment. In this experiment, we have a spring hanging from a clamp, and we add weights to the spring. We measure the original length and then measure the length of the spring after adding weights increasing by 50g (0.05 kg) each time and record the lengths, total masses and the force applied by the masses. (We take gravity to be 10 N/kg)

We calculate the force by multiplying the mass by acceleration due to gravity, and we find extension by subtracting the original length (length with no mass added) from the length after mass is added.

When we plot this on a graph of extension versus force, we can see additional information:

As you can see, extension is directly proportional to force (the graph is a straight line from the origin or 'zero point'), until the point known as the deformation point or elastic limit. At this point, the spring will no longer return to its original length once the masses have been removed. The spring is no longer experiencing elastic deformation, but is rather experiencing plastic deformation.

If we were to find the slope of an extension vs force graph:

We would end up with a value in m/N. If you remember from previously when we stated that the spring constant is measured in N/m, you will realize that this is simply the reciprocal or inverse of the spring constant, k⁻¹. So, if you wanted to find the spring constant from a graph, you would do this:

Example question- How much force is needed to pull a spring with a spring constant of 20 N/m a distance of 25 cm?

F = k × e

= 20 N/m × 0.25m

= 5 N

(You can find more here)

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