First slide

Elastic Modulie

Question

Wires A and B are connected with blocks P and Q as shown. The ratio of lengths, radii and Young's modulus of wires A and B are r:1, 1:2r and 1:3r respectively (r is a constant). Find the mass of block P if ratio of increase in their corresponding lengths is $\frac{1}{6 r^{2}}$ . The mass of the block Q is 3M.

Moderate

A

M
B

3M
C

6M
D

9M

Solution

$∆ l = \frac{FL}{AY}$
$\frac{{(∆ l)}_{A}}{{(∆ l)}_{B}} = (\frac{F_{A}}{F_{B}}) (\frac{L_{A}}{L_{B}}) (\frac{A_{B}}{A_{A}}) (\frac{Y_{B}}{Y_{A}})$
= $(\frac{KM}{3 M}) (r) {(\frac{1}{2 r})}^{2} (\frac{1}{3 r}) = \frac{K}{36 r^{2}} = \frac{1}{6 r^{2}} (given)$
K = 6
Mass of block P = 6 M

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