Force 6xtension Graphs

Force 6xtension Graph Basics

A force 6xtension graph plots the force applied to an object (usually a spring) on the vertical axis (in newtons, N) against the extension of the object on the horizontal axis (in metres, m). Extension means how much the object stretches from its original length.

In the initial part of the graph, the line is straight and passes through the origin, showing that force and extension are directly proportional. This straight part is called the linear region.

The gradient (steepness) of the straight line gives information about the stiffness of the spring or material. A steeper gradient means a stiffer spring that requires more force to produce the same extension.

For example, if a spring stretches 0.02 m when a force of 4 N is applied, the gradient is:

$\text{Gradient} = \frac{\text{Force}}{\text{Extension}} = \frac{4\, \text{N}}{0.02\, \text{m}} = 200\, \text{N/m}$

Hooke's Law

Hooke's Law states that the force applied to stretch or compress a spring is directly proportional to the extension or compression produced, as long as the elastic limit is not exceeded.

This can be written as the formula:

$$F = k \times e$$

• $F$ is the force applied (in newtons, N)
• $e$ is the extension of the spring (in metres, m)
• $k$ is the spring constant or stiffness (in newtons per metre, N/m)

The spring constant $k$ is a measure of how stiff the spring is: a larger $k$ means a stiffer spring.

The limit of proportionality is the point on the graph where the force and extension stop being proportional. Beyond this point, the graph curves and Hooke 2s Law no longer applies.

For instance, if a spring extends by 0.05 m under a force of 10 N, the spring constant is:

$$k = \frac{F}{e} = \frac{10\, \text{N}}{0.05\, \text{m}} = 200\, \text{N/m}$$

Elastic and Plastic Deformation

The force 6xtension graph shows two main regions:

• Elastic region: The initial straight part of the graph where the material returns to its original shape when the force is removed. Deformation is reversible here.
• Plastic region: Beyond the limit of proportionality, the graph curves and the material undergoes permanent deformation. The shape does not fully return to normal when the force is removed.

The curve beyond the limit of proportionality shows that the material is being stretched beyond its elastic limit, leading to plastic deformation.

For example, a metal wire stretched gently will return to its original length (elastic), but if stretched too far, it will be permanently longer (plastic).

Interpreting Graphs

You can use force 6xtension graphs to:

• Calculate the spring constant $k$: The gradient of the linear part of the graph is equal to $k$. Calculate it by dividing force by extension in the straight region.
• Identify the limit of proportionality: The point where the graph first starts to curve away from a straight line.
• Estimate energy stored: The area under the force 6xtension graph represents the work done to stretch the spring, which is the energy stored (elastic potential energy). (See separate notes on elastic potential energy.) A brief formula is: Energy stored = 0.5 6 force 6 extension in the linear region.

For example, if the linear part of a graph shows a force of 8 N at an extension of 0.04 m, then:

$$k = \frac{8\, \text{N}}{0.04\, \text{m}} = 200\, \text{N/m}$$