Question

6. A composite rod is made by joining two rods of different materials but with the same cross-sectional area as shown below. Design such a composite rod using steel (α = 11 x 10-6 / C°) and brass (α = 19 x 10-6 / C°) whose total length is 52.4 cm and whose effective coefficients of linear expansion is 13 x 10-6 / C°. Compute the lengths of the steel and brass rods that must be used. Neglect changes in cross-sectional area of the rods.

Answer 1

Given data ,

1. coefficients of linear expansion of steel α1 = 11 x 10^-6 C°
2. coefficients of linear expansion of brass α2 = 19 x 10^-6 / C°
3. whose total length L = 52.4 cm
4. whose effective coefficients of linear expansion is α = 13 x 10^-6 / C°
5. neglect changes in cross sectional area of rods

To. find : the lengths of the steel(L1) and brass(L2) rods that must be used

Solution :

formal to be used :

1. L = L1 + L2 = 52.4 cm .....(1)
2. $L1 = \frac{L( \alpha - \alpha 2)}{( \alpha 1 - \alpha 2)} \: .......(2)$
3. $L2 = \frac{L( \alpha - \alpha 1)}{( \alpha 2 - \alpha 1)} \: ........(3)$

putting value in equation 2 we get ,

$L1 = \frac{54.4(13 \times {10}^{ - 6} - 19 \times {10}^{ - 6}) }{11 \times {10}^{ - 6} - 19 \times {10}^{ - 6} } = 40.8 \: m$

on putting the value of L1 in equation 1 we get ,

$54.4 = L2 + 40.8$

$L2 = 54.4 - 40.8 = 13.6 \: m$

Hence the lengths of the steel(L1) and brass(L2) rods that must be used is 40.8 m and 13 .6 m

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