Question

Which of the following statements is correct?
(a) Hooke’s Law is applicable only within the elastic limit.
(b) The adiabatic and isothermal elastic constants of a gas are equal.
(c) Young’s Modulus has no dimensions.
(d) Stress multiplied by strain is equal to the stored energy.

Answer 1

Hint:We will understand Hooke's Law briefly here with the help of stress and strain relation and then search for the options mentioned here to get the correct statements. We will know in brief about the elastic limit and hence find whether statement (a) is correct or not. For statement (b) we know that both these constants are different from the solid object’s aspect, it is for gases. Then we will find the dimensional formula of Young’s modulus and hence find the statement (c) is correct or incorrect. For, statement (d) we know that half of stress multiplied by strain is equal to the stored energy. So, it is incorrect.

Complete step by step answer:
Let us first consider the statement (a). When the stress and strain are small enough, we often find that the two are directly proportional and we call the proportionality constant and elastic modulus. The harder you pull on something, the more it stretches, the more you squeeze it, the more it compresses. We can express this relationship as an equation,
$\dfrac{{{\text{Stress}}}}{{{\text{Strain}}}} = {\text{Elastic Modulus (Hooke's Law)}}$

Hooke’s Law only works upto a certain limit after which the stress-strain graph does not follow the general pattern. The limit till which the Hooke’s Law is valid is known as elastic limit.So, statement (a) Hooke’s Law is applicable only within the elastic limit is correct.

Let us now consider the statement (b). The adiabatic elastic constants of a gas is different from the isothermal elastic constants of a gas. So, the statement (b) is incorrect. Let us consider now the statement (c). The dimension of Young’s Modulus is $M{L^{ - 1}}{T^{ - 2}}$. So, the statement (c) is incorrect. Let us consider now the statement (d). The half of stress multiplied by strain is elastic stored energy per unit volume. So, statement (d) is also incorrect.

Therefore, the correct option is A.

Note:It must be noted that stress is defined as the force that acts on the object per unit area and strain is defined as the relative change in the dimensions of the body subjected to the force acting on it. The restoring force is defined as the opposing force which tries to compensate for the force acting on it, and restore its original shape.