Elasticity can be defined as the ability of a material to return to its original shape and size upon the removal of a load. For example, when I placed my fingers on both sides of a rubber band and expand it, it is observed to get expanded, upon removing my hand from the rubber band and it returns to it's original shape and size, it is said to have undergone elasticity. If an elastic material can no longer regain its original shape and size, such phenomenon is called plasticity.
The graph above shows the different stages of an elastic material stretched by a force.
Where F1 represent first force, F2 represent second force, e1 represent first extension, e2 represent second extension, K represent force constant or elastic constant in N/m.
- Elastic limit (E): this is the maximum load (force) which a body can experience and still retain its original shape and size once the load has been removed.
- Yield point (Y): this is reached when a stretched wire does not return to its original position.
- Maximum load(M): this is the maximum load (force) the body can sustain before it breaks.
- Breaking point(B): this is the point at which the wire breaks.
This law states that the force applied to an elastic material is directly proportional to its extension provided that the elastic limit is not exceeded. What it means is that, when you apply little force, it produces little extension and a bigger force will produce greater extension as long as the elastic limit is not exceeded. Hooke's law can be expressed mathematically as
Where F1 represent first force, F2 represent second force, e1 represent first extension, e2 represent second extension, K represent force constant or elastic constant in N/m.
e1 = L1 – L0 while e2 = L2 – L0 substitute in equation 2
Making L0 the subject of the formular, you have
Where L0 is called the original/natural length
Stress is the force acting on the unit area of a material. It is expressed in N/m²
Strain is defined as the ratio of extension to the original length. Strain on a body is as a result of stress. It has no unit.
Young's modulus is the measure of how stiff a material is. It is the ratio of stress to strain. Young's modulus has the same unit as stress. It can be expressed mathematically as
Young's modulus can also be expressed as 
Where K is the elastic constant in N/m, l0 is the original length, A is the area in m² and F is the force in N.
When work is done on an elastic body (spring), energy is stored in that spring. The energy stored in that spring can be calculated using several formulae like:
Examples
- If a load of 5kg stretches a cord by 1.5cm, what is the force constant of the cord? (g= 10m/s^2).
- A force of 200N is applied to a steel wire of cross sectional area 0.4m², the stress is?
Solution
- If a force of 40N stretches a wire from 30m to 30.02m, what is the amount of force required to stretch the same material from 30m to 30.05m?
Solution
- If a wire 40cm long is extended to 40.5cm by a force of 400N. Find the strain energy of the wire.
Solution
Questions
- An elastic material has a length of 36cm when a load of 40N is hung on it and a length of 45cm when a load of 60N is hung on it. The original length of the string is?
- A spring of force constant 400N/m is compressed such that its length shortens by 4cm. The energy stored in the spring is?
- An elastic string of length 10cm extends to 50cm when it supports a weight of 100N. Find the energy stored.
- A force of 15N stretches a spring to a total length of 30cm. an additional force of 10N stretches the spring 5cm further. Find the natural length of the spring.
- If the stress on a wire is 5 * 10⁷N/m² and the wire stretched from its original length of 10cm to 30cm. find the young's modulus of the wire.