Hooke's Law (Cambridge International A-Level Physics)

Spec point — 6.1.3

Hooke's law

• A material demonstrating elastic behaviour obeys Hooke’s Law if its extension is directly proportional to the applied force (load)
• The Force-extension graph of an object obeying Hooke's law is a straight line through the origin
• This linear relationship is represented by the Hooke’s law equation

$F=kx$

• Where:

  • F is force applied in N
  • k is the spring constant in N m⁻¹
  • x is the extension of the spring

The Spring Constant

Spec point — 6.1.4

• k is the spring constant of the spring and is a measure of the stiffness of a spring

  • A stiffer spring will have a larger value of k

• k is defined as the force per unit extension up to the limit of proportionality

• The SI unit for the spring constant is N m^-1

• Rearranging the Hooke’s law equation shows the equation for the spring constant is

$k=\frac{F}{x}$

• Therefore, the spring constant k is the gradient of the linear part of a force-extension graph

Gradient of force-extension graph

Spring constant is the gradient of a force vs extension graph

Worked Example A spring was stretched with increasing load.

The graph of the results is shown below.

Determine the value of the spring constant.

Answer:

Step 1: Rearrange Hooke's Law:

• Spring constant, k, is:

$k=\frac{F}{x}$

Step 2: Relate the gradient of this graph to k :

• The y axis of this graph is length L and the x axis is load F
• Gradient is change in y over change in x:

$\mathrm{gradient}=\frac{ΔL}{ΔF}=\frac{x}{F}$

• The change in length is the extension x
• Therefore:

$\mathrm{gradient}=\frac{1}{k}$

Step 3: Determine the gradient of the graph:

• Choose a large section of the graph line to determine the changes in the x and y axes

• Convert the extension from cm to m

$\mathrm{gradient}=\frac{0.145-0.100}{0.36}=0.125mN^{-1}$

Step 4: Calculate the spring constant:

• The spring constant is

$k=\frac{1}{\mathrm{gradient}}=\frac{1}{0.125}=8.0Nm^{-1}$

Exam Hint Double check the axes before finding the spring constant as the gradient of a force-extension graph. Exam questions often swap the load onto the x-axis and length on the y-axis. In this case, the spring constant is $\frac{1}{\mathrm{gradient}}$ and not the spring constant.