First slide

Elastic Modulie

Question

The radii and Young's moduli of two uniform wires A and B are in the ratio 2 : 1 and 1 : 2 respectively. Both wires are subjected to the same longitudinal force. If the increase in length of the wire A is one percent, the percentage increase in length of the wire B is

Moderate

A

1.0
B

1.5
C

2.0
D

3.0

Solution

Young's modulus, $Y = \frac{FL}{{πr}^{2} ∆ L}$
or $\frac{∆ L}{L} = \frac{F}{{πr}^{2} Y}$
For the same force,
$\frac{\frac{∆ L_{B}}{L_{B}}}{\frac{∆ L_{A}}{L_{A}}} = {(\frac{r_{A}}{r_{B}})}^{2} (\frac{Y_{A}}{Y_{B}})$
$\frac{∆ L_{B}}{L_{B}} = 2 \frac{∆ L_{A}}{L_{A}}$
$\frac{∆ L_{B}}{L_{B}} \times 100 = 2 (\frac{∆ L_{A}}{L_{A}} \times 100) = 2 %$

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