Hooke’s law states that “the extension of a spring is proportional to the applied force, provided that the force is not large enough to deform the spring permanently”. Mathematically expressed as Force α extension.

## Spring constant

Since Force α extension then Force / Extension = constant (k). The constant of proportionality (k) is called the spring constant. F / e = k or Force (N) = k e. The spring constant is a measure of the stiffness of a spring. The greater the constant the stiffer the spring.

The spring constant varies with the following;-

- a) Material – identical springs mad of different materials will have different constants i.e.steel and copper.
- b) Diameter – the stiffness decreases with the increase in diameter.
- c) Thickness of the wire – a spring made of a thicker wire is stiffer than the one made of thin wire of the same material.
- d) Length of spring – a short spring is stiffer than a longer one.
- e) Number of turns per unit length – a spring with higher number of turns per unit length is less stiff than the one with fewer turns per unit length.

Example

- If the springs shown below are similar and the constant of proportionality (k) is 100 Nm
^{-1}, determine total extension in each arrangement.

## The spring balance

It is made up of a spring mounted in a metal or plastic casing. The spring is fitted with a pointer which moves along a calibrated scale divided into ten equal parts.

Examples

- A load of 4 N causes a certain copper wire to extend by 1.0 mm. Find the load that will cause a 3.2 mm extension on the same wire. (Assume Hooke’s law is obeyed).

Solution

F α e also F_{1} / F_{2} = e_{1} / e_{2} = F_{2} = (4 × 3.2) / 1.0 = 12.8 N. 2. A body of 200 g was hung from the lower end of a spring which obeys Hooke’s law. Given that the spring extended by 100 mm, what is the spring constant for this spring? Solution F = α e, F = k e. F = 200 × 10-3 kg × 10 N /kg = 2 N. Extension = 100 × 10^{3} m = 0.1 m.

Spring constant (k) = 2 / 0.1 = 20 N/m.

- Two identical springs, whose spring constant is 6.0 N/cm, are used to support a load of 60 N as shown below. Determine the extension of each spring.

Solution

Since the springs are parallel their spring constant equals 2k. Therefore extension = Force / k = 2 F / k = 60 / 2 × 6 = 5 cm. Each spring will extend by 5 cm.