Force and extension relationship

The spring constant measures the stiffness of the spring. A higher spring constant indicates a stiffer spring that requires a larger force to stretch it by the same amount.

Worked Example: Calculating Spring Constant

A metal spring has an original length of $0.15\,\text{m}$. When a mass with a weight of $8.0\,\text{N}$ is hung from it, the new length is $0.25\,\text{m}$. Calculate the spring constant.

Step 1: Calculate the extension.

• $e = 0.25\,\text{m} - 0.15\,\text{m} = 0.10\,\text{m}$

Step 2: Rearrange the Hooke's Law equation for $k$.

• $k = \frac{F}{e}$

Step 3: Substitute the values and calculate.

• $k = \frac{8.0}{0.10}$
• $k = 80\,\text{N/m}$

Elastic and Inelastic Deformation

If you gently stretch a hair tie, it snaps back to its original size. This is elastic deformation, where the object completely returns to its original shape and length once the stretching force is removed.

However, if you pull too hard on a metal spring, it stretches permanently. This permanent change in shape is called inelastic deformation (often referred to as plastic deformation, though OCR Gateway specifically prefers the term "inelastic").

This permanent deformation occurs when the applied force exceeds the elastic limit of the material. If you unload the spring after passing this point, the unloading line on a graph will be parallel to the initial loading line, but it will not return to zero.

Analysing Force-Extension Graphs

We can plot the force applied against the extension to visualise how a material behaves.

• The Linear Region: For a spring obeying Hooke's Law, the graph starts as a straight line passing through the origin $(0,0)$. The gradient (steepness) of this straight line represents the spring constant ($k$).
• The Non-Linear Region: Eventually, the line begins to curve. The exact point where it stops being a straight line is the limit of proportionality. Past this point, the gradient decreases, meaning a smaller increase in force causes a much larger extension.
• Work Done and Energy: Stretching a spring requires work. This work is stored as elastic potential energy ($E_e$). You can calculate this stored energy by calculating the total area under the force-extension graph.

Other Simple Systems

Not all materials obey Hooke's Law like a metal spring does.

• Rubber bands: These do not have a linear proportional region. They produce a non-linear, "S-shaped" curve from the very beginning. The loading and unloading curves do not perfectly match, creating a loop due to hysteresis, where energy is dissipated as heat.
• Polythene strips: These undergo plastic flow almost immediately. They stretch easily and deform permanently, with no significant linear proportional region. The work done is used to permanently rearrange the molecular structure.

Investigating Force and Extension (PAG 02)

To investigate Hooke's Law safely and accurately, suspend a spring from a clamp stand. Crucially, use a G-clamp to secure the stand to the bench so it does not topple over when heavy weights are added.

Attach a pointer, also called a fiducial marker, to the bottom of the spring. Read the extension against a vertical metre ruler. Using a pointer helps ensure your eye is exactly level with the reading, significantly reducing parallax error.