Question

Two wires $A$ and $B$ of radius $2r$ and $r$ respectively are joined and a force $F$ is applied at the end. The length of each wire is $L$. Their Youngs modulus are Y and 2Y respectively. Find the net elongation.

Answer 1

Given.

Length of both wires, $L$

Radius of both wires, $r_A=2r_B=2r$

Young modulus of A and B, $2Y_A=Y_B=2Y$

Total extension of joined wire, $\Delta L_{net}$

Area of B$=\pi r^2=A$

Area of A $=\pi {\left(2r\right)}^2=4\pi r^2=4A$

Both wires join in series, so tension in both is equal

$T=\frac{Y_A{A}_A\Delta L_A}{L}=\frac{Y_B{A}_B\Delta L_B}{L}$

Net extension in joined wire.

$\Rightarrow \Delta L_{net}=\Delta L_A+\Delta L_B=\frac{TL}{Y_A{A}_A}+\frac{TL}{Y_B{A}_B}=\frac{TL}{Y\left(4A\right)}+\frac{TL}{\left(2Y\right)A}$

$\Rightarrow \Delta L_{net}=0.75\frac{TL}{YA}$

Net extension is $0.75\frac{TL}{Y\pi r^2}$