First slide

Stress and strain

Question

If the ratio of diameters, lengths and Young's modulus of steel and copper wires shown in the figure are p, q and s respectively, then the corresponding ratio of increase in their lengths would be

Moderate

A

$\frac{5 q}{(7 s p^{2})}$
B

$\frac{7 q}{(5 s p^{2})}$
C

$\frac{2 q}{(5 s p)}$
D

$\frac{7 q}{(5 s p)}$

Solution

$As Y = \frac{F L}{A Δ L} = \frac{4 F L}{π D^{2} Δ L} Δ L = \frac{4 F L}{π D^{2} Y} ∴ \frac{Δ L_{S}}{Δ L_{C}} = \frac{F_{S}}{F_{C}} \frac{L_{S}}{L_{C}} \frac{D_{C}^{2}}{D_{S}^{2}} \frac{Y_{C}}{Y_{S}}$
$where subscripts S and C refer to copper and steel respectively. H e r e, F_{S} = (5 m + 2 m) g = 7 m g F_{C} = 5 m g \frac{L_{S}}{L_{C}} = q, \frac{D_{S}}{D_{C}} = p, \frac{Y_{S}}{Y_{C}} = s ∴ \frac{Δ L_{S}}{Δ L_{C}} = (\frac{7 m g}{5 m g}) (q) {(\frac{1}{p})}^{2} (\frac{1}{s}) = \frac{7 q}{5 p^{2} s}$

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