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Hooke’s Law

Hooke’s Law  is  a law named after 17th century British physicist Robert Hooke, who sought to demonstrate the relationship between the forces applied to a spring and its elasticity.

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

 F= -kX.

Where

F  is the force applied to the spring (either in the form of strain or stress)

-X is the displacement of the spring, with a negative value demonstrating that the displacement of the spring once it is stretched;

And

 k is the spring constant, which is a measure of the stiffness of a spring.and details just how stiff it is.

Illustration of Hooke’s Law, showing the relationship between force and distance when applied to a spring.

The spring constant varies with the following;-

1. a) Material – identical springs mad of different materials will have different constants i.e.steel and copper.
2. b) Diameter – the stiffness decreases with the increase in diameter.
3. 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.
4. d) Length of spring – a short spring is stiffer than a longer one.
5. 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.

Hooke’s law is a classical example of an explanation of elasticity—which is the property of an object or material which causes it to be restored to its original shape after distortion.

This ability to return to a normal shape after experiencing distortion can be referred to as a “restoring force”. Understood in terms of Hooke’s Law, this restoring force is generally proportional to the amount of “stretch” experienced.

In addition to governing the behavior of springs, Hooke’s Law also applies in many other situations where an elastic body is deformed. These can include anything from inflating a balloon and pulling on a rubber band to measuring the amount of wind force is needed to make a tall building bend and sway.

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.

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Examples

1. 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 F₁ / F₂ = e₁ / e₂ 

= F₂ = (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 × 10³ m = 0.1 m.

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

3. 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.