Question

A rigid bar of mass $15kg$ is supported symmetrically by three wires each $2.0m$ long. Those at each end are of copper and the middle one is of iron. Determine the ratios of their diameters if each is to have the same tension.

Answer 1

Young's modulus of copper $\left(Y1\right)=110\times {10}^9\frac{N}{m^2}$

Young's modulus of steel $\left(Y2\right)=190\times {10}^9\frac{N}{m^2}$

Let D1 and D2 have diametres of copper and steel respectively.

A/C to question ,

Tension are same for both

Youngs modulus $=\frac{stress}{strain}$

$Y=\frac{F}{A}\left(\frac{\Delta L}{L}\right)$

$Y=\frac{FL}{\frac{\pi d^2}{4}\times \Delta L}$

Y is inversely proportional to square of diametre .
So, $\frac{Y1}{Y2}=\frac{d2²}{d1²}$

$\frac{d1}{d2}=\sqrt{\frac{Y2}{Y1}}$

$=\sqrt{190\times {10}^9/110\times {10}^9}=\sqrt{\frac{9}{11}}=1.31$

$d1:d2=1.31:1$