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. $\frac{Y_1{Y}_{2}A}{2(Y_1{L}_2+Y_2{L}_1)}$
• B. $\frac{Y_1{Y}_{2}A}{2(L_1{L}_2{)}^{1/2}}$
• C. $\frac{Y_1{Y}_{2}A}{(Y_1{L}_2+Y_2{L}_1)}$
• D. $\frac{(Y_1{Y}_2{)}^{1/2}A}{2(L_1{L}_2{)}^{1/2}}$

Answer 1

The correct option is C $\frac{Y_1{Y}_{2}A}{(Y_1{L}_2+Y_2{L}_1)}$

The wires are connected in series

$Y=\frac{Fl}{A\Delta l}\Rightarrow \frac{YA}{l}=\frac{F}{\Delta l}\Rightarrow k=\frac{YA}{l}$

$K$ is spring constant

In series spring constants are added as

$\frac{1}{K_1}+\frac{1}{K_2}=\frac{1}{K}\Rightarrow \frac{l_1}{Y_{1}A}+\frac{l_2}{Y_{2}A}=\frac{1}{K}$

$\Rightarrow K=\frac{Y_1{Y}_{2}A}{Y_1{L}_2+Y_2{L}_1}$