Question

A metal wire of length $L_1$​ and area of cross section A is attached to a rigid support. Another metal wire of length $L_2$​ and of the same cross-sectional area is attached to the free end of the first wire. A body of mass M is then suspended from the free end of the second wire. If $Y_1$​ and $Y_2$​ are the Young's moduli of the wires respectively, the effective force constant of the system of two wires is
A: $\dfrac{[({{Y}_{1}}{{Y}_{2}})A]}{[2({{Y}_{1}}{{L}_{2}}+{{Y}_{2}}{{L}_{1}})]}$
B: $\dfrac{[({{Y}_{1}}{{Y}_{2}})A]}{{{({{L}_{1}}{{L}_{2}})}^{1/2}}}$
C: $\dfrac{[({{Y}_{1}}{{Y}_{2}})A]}{[({{Y}_{1}}{{L}_{2}}+{{Y}_{2}}{{L}_{1}})]}$
D: $\dfrac{[{{({{Y}_{1}}{{Y}_{2}})}^{1/2}}A]}{{{({{L}_{1}}{{L}_{2}})}^{1/2}}}$

Answer 1

Hint: When two heavy wires are connected like a chain and hung up from a height with an attached mass, it experiences an elongation due to elastic force. It experiences a strain and the length is changed. This might also lead to an oscillation from the original length to the stretched length and vice versa.
Formula used:
Force constant $K=\dfrac{F}{\Delta L}$, where F is the force acting on the body and $\Delta L$is the change in length of the body.
\[{{k}_{eq}}=\dfrac{{{k}_{1}}{{k}_{2}}}{{{k}_{1}}+{{k}_{2}}}\]: The formula to find the effective force constant of two bodies connected in series.

Complete step by step answer:
We’re given the lengths of two wires as $L_1$ and $L_2$. Young's moduli are $Y_1$ and $Y_2$. They have the same area of cross section A.
We know that the force constant $K=\dfrac{F}{\Delta L}=\dfrac{YA}{L}$ (Since $Y=\dfrac{FL}{A\Delta L}$)
\[{{k}_{eq}}=\dfrac{{{k}_{1}}{{k}_{2}}}{{{k}_{1}}+{{k}_{2}}}\] is the formula to find the effective force constant of two bodies connected in series.

Substituting the corresponding values of K, we get
\[{{k}_{eq}}=\dfrac{(\dfrac{{{Y}_{1}}A}{{{L}_{1}}})(\dfrac{{{Y}_{2}}A}{{{L}_{2}}})}{(\dfrac{{{Y}_{1}}A}{{{L}_{1}}})+(\dfrac{{{Y}_{2}}A}{{{L}_{2}}})}=\dfrac{({{Y}_{1}}{{Y}_{2}})A}{{{Y}_{1}}{{L}_{2}}+{{Y}_{2}}{{L}_{1}}}\]

So, the correct answer is “Option C”.

Note:
Young’s modulus is used to determine rigidity of a body .It is also used to determine the level of deformation of a body in a given load. Lower the value of Young’s modulus, more deformation is experienced by the body.