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Q.

A cylindrical pillar is made of two materials. The inner core, having radius $r,$ has Young’s modulus $Y_{1}$ and the outer layer (between radius $r$ and $R$ ) has Young’s modulus $Y_{2} .$ A load of weight $W$ is placed on the cylinder. Find the fraction of load supported by the outer layer. $[R = 3 r, \frac{Y_{1}}{Y_{2}} = 2]$

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Detailed Solution

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$A_{1} = π r^{2}; A_{2} = π (R^{2} - r^{2})$
$σ_{1} A_{1} + σ_{2} A_{2} = W ............ (i)$
Strain in both sections is same.
$\frac{σ_{1}}{Y_{1}} = \frac{σ_{2}}{Y_{2}} ............ (i i)$
Solving (i) and (ii) gives $σ_{2} = \frac{W Y_{2}}{A_{1} Y_{1} + A_{2} Y_{2}}$
Required fraction is: $\frac{σ_{2} A_{2}}{W} = \frac{A_{2} Y_{2}}{A_{1} Y_{1} + A_{2} Y_{2}}$
$= \frac{(R^{2} - r^{2}) Y_{2}}{r^{2} Y_{1} + (R^{2} - r^{2}) Y_{2}}$

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