Concept of Elasticity:
Solids tend to change their size and shape whenever sufficiently strong external forces are exerted to them and to return to their original size or shape after the forces causing the change are removed. The Solids that retain their shape or size after the force causing the change has been removed are stated to be 'elastic', and this property of solids is termed as elasticity.
1) Elasticity is the capability of a substance to get back its original size and shape after being distorted through an external force.
2) An elastic material is one which regains its original size and shape after distorting the external force which has been removed.
Statement of Hooke's Law:
Hooke's law defines that, given the elastic limit of an elastic material is not exceeded, the extension, 'e', of the material is directly proportional to the applied force, F.
F ∝ e
That is, F = K e
Here, k is the constant of proportionality termed as elastic constant or force constant or stiffness of the material.
From the above formula, K = F/e
If 'F' is in Newton and 'e' in meters, then K is in Newton per metre (Nm-1)
Elastic constant or stiffness of the elastic material is the force needed to produce the unit extension of the material.
The working of spring balance is dependent on Hooke's law. In this situation F = mg, the weight of the body that is proportional to 'e', the extension of the spring.
Experimental Verification of Hooke's Law:
Consider two similar metallic wires A and B on which main scale and vernier scales are fixed and the wires are hanged from the rigid support.
The kinks generated on the reference wire A and experimental wire B are eradicated by loading weight at their free ends termed as dead loads by the assistance of meter scale length l at wire B is measured and by the assistance of micrometer screw gauge its radius 'r' is as well measured.
Now, main scale reading and vernier scale readings are noted. The equivalent loads are added on the pan of wire B and corresponding reading are noted. Assume, w1, w2, w3 and w4 are the weights on the wire B and e1, e2, e3 and e4 are corresponding elongations generated, in elastic limit.
As, Y = (F/A)/(Δl/l) = (Fl)/(πr2e)
When a graph is plotted between F and e, a straight line from horizon is achieved whose slope F/e is computed and we have,
Y = Slope x (l/πr2)
Then, the ration (weight/Elongation) is computed and found that, F/e = constant
That is, F/e = constant
Thus, F ∝ e
Young's Modulus of Elasticity:
Assume that a wire of length l (m) and cross-sectional area A (m2) is extended via e (m) through a force F (N).
(i) The ratio of the force to area, F/A is termed as the stress or 'tensile' of the elastic material.
Stress = F/A
(ii) The ratio of extension, 'e' to the original length, l of the wire that is, e/l is termed as the tensile strain of the wire.
Therefore Strain = e/l
F = stress x A
e = Strain x l
By using Hooke's law, F = ke
∴ Stress x A = k x Strain x l
∴ Stress = (kl)/A x strain
∴ Stress = k1
Constant = kl/A
Stress/Strain = k1
Stress ∝ Strain
Therefore Hooke's law can as well be stated as follows:
The tensile stress of the material is directly proportional to the tensile strain given the elastic limit is not surpassed.
The constant of proportionality, k1 is termed as Young's modulus of elasticity and is symbolized by the symbol 'γ'.
∴ Young's modulus (γ) = Stress/Strain
(γ) = (F/A)/(e/l)
The unit of γ is Nm-2 (Newton per square metre) the similar unit as stress, as strain consists of no unit.
Dimension of γ = (Dimension of stress)/(Dimension of strain)
Elastic Potential Energy:
Definition: The elastic potential energy of a compressed or stretched material is the capability of the material to do work.
Elastic potential energy occurs due to work done in stretching or compressing the material.
W = 1/2 Fe = (1/2) ke2
Here 'F' is the maximum stretching (or compressing) force, 'e' is the extension (or compression) and 'k' is the force constant or rigidity of the material.
Illustration or application of elastic potential energy:
Whenever you stretch the rubber of a catapult and project a stone, the elastic potential energy stored in the rubber is transformed into the kinetic energy of the flying stone according to the law of conservation of energy.
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