## Hooke's Law: Introduction

- 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**then**

*Force α extension*

*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;

**Material**– identical springs mad of different materials will have different constants i.e. steel and copper.**Diameter**– the stiffness decreases with the increase in diameter.**Thickness of the wire**– a spring made of a thicker wire is stiffer than the one made of thin wire of the same material.**Length of spring**– a short spring is stiffer than a longer one.**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.

#### Practice Example 1

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

##### Solution

k = 100 Nm^{-1}

extension = force / k = 10 ÷ 100 = 0.1 m = 10 cm. Extension of the lower spring = 10 cm, extension of the two parallel springs = 5 cm.

Total extension = 10 + 5 = 15 cm. Extension of the two lower springs = 5 cm

Middle spring extend by = 10 cm

Upper two springs extend by = 5 cm

Total extension = 5 + 10 + 5 = 20 cm.