Hooke's Law

A material obeys Hooke’s Law if its extension is directly proportional to the applied force (load)
The Force v Extension graph is a straight line through the origin (see “Extension and Compression”)
This linear relationship is represented by the Hooke’s law equation

$F = k x$

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

Worked example

A spring was stretched with increasing load.

The graph of the results is shown below.

WE - hookes law question image, downloadable AS & A Level Physics revision notes

What is 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:

$gradient = \frac{Δ L}{Δ F} = \frac{x}{F}$

where change in length is just extension x

Therefore

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

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

6-1-2-we-hookes-law-cie-new

Convert the extension from cm to m

$gradient = \frac{0.145 - 0.100}{0.36} = 0.125 m N^{- 1}$

Step 4: Calculate the spring constant:

The spring constant is

$k = \frac{1}{gradient} = \frac{1}{0.125} = 8.0 N m^{- 1}$

Exam Tip

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 gradient is not the spring constant but 1 ÷ gradient is.

The Spring Constant

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

It is defined as the force per unit extension up to the limit of proportionality (after which the material will not obey Hooke’s law)
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

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

The spring constant is the force per unit extension up to the limit of proportionality (after which the material will not obey Hooke’s law)
Therefore, the spring constant k is the gradient of the linear part of a Force v Extension graph

Force against Extension Graph

Spring constant on graph, downloadable AS & A Level Physics revision notes

Spring constant is the gradient of a force vs extension graph

Combination of springs

Springs can be combined in different ways
- In series (end-to-end)
- In parallel (side-by-side)
Springs in Series and Parallel

Series and parallel springs, downloadable AS & A Level Physics revision notes

Springs combined in parallel have a greater equivalent spring constant, springs combined in series have a lower equivalent spring constant.

This is assuming k₁ and k₂ are different spring constants
The equivalent spring constant for combined springs are summed up in different ways depending on whether they’re connected in parallel or series

Worked example

Three springs are arranged vertically as shown. Springs P, Q and O are identical and have spring constant k . Spring R has spring constant 4k . What is the increase in the overall length of the arrangement when a force W is applied as shown?

A. $\frac{12 k}{7 W}$

B. $\frac{6 W}{5 k}$

C. $\frac{7 W}{12 k}$

D. $\frac{2 W}{5}$

Answer: C

Step 1: Find the equivalent spring constant of springs O, P and Q:

As O, P and Q are in parallel, they act as one spring with a larger spring constant, k_OPQ
Recall that the spring constants are added for parallel springs, and all three have spring constant k

$k_{O P Q} = k_{O} + k_{P} + k_{Q} = 3 k$

Step 2: Include spring R to find the total spring constant:

The system is now effectively two springs in series
- The three parallel springs can be considered as one spring with spring constant 3k
To find total spring constant, k_OPQR , recall the equation for spring constant when combining springs in series

$\frac{1}{k_{O P Q R}} = \frac{1}{k_{R}} + \frac{1}{k_{O P Q}} = \frac{1}{4 k} + \frac{1}{3 k} = \frac{7}{12 k}$

$k_{O P Q R} = \frac{12 k}{7}$

Step 3: Use Hooke's law to determine extension:

Now we can treat the system as one spring with constant k_OPQR and apply Hooke's law to find extension, x , when load W is applied

$x = \frac{W}{k_{O P Q R}} = \frac{W}{(\frac{12 k}{7})} = \frac{7 W}{12 k}$

Exam Tip

The equivalent (or effective) spring constant equations for combined springs work for any number of springs e.g. if there are 3 springs in parallel k₁ , k₂ and k₃ , the equivalent spring constant would be k_eq = k₁ + k₂ + k₃ .

Hooke's Law (CIE A Level Physics)

Revision Note

Hooke's Law

Worked example

Exam Tip

The Spring Constant

Force against Extension Graph

Combination of springs

Springs in Series and Parallel

Worked example

Exam Tip

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Author: Katie M