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Question

The coefficient of linear expansion of steel and brass are 11×106/C and 19×106/C, respectively. If their differences in lengths at all temperatures has to kept constant at 30 cm, their lengths at 0C should be

A
71.25 cm and 41.25 cm
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B
82 cm and 52 cm
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C
92 cm and 62 cm
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D
62.25 cm and 32.25 cm
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Solution

The correct option is A 71.25 cm and 41.25 cm
Let us suppose Lo1 and Lo2 is the length of rod of steel and brass at 0C respectively and ΔT change in temperature.
We know, L=Lo(1+αΔT) [linear expansion]
For steel,
L1=Lo1(1+11×106ΔT) [αsteel=11×106/C,given]
=Lo1+11×106×ΔT×Lo1...(1)
For brass,
L2=Lo2(1+19×106ΔT) [αbrass=19×106/C,given]
=Lo2+19×106×ΔT×Lo2...(2)
On subtracting (1) from (2)
L2L1=Lo2Lo1+19×106×ΔT×Lo211×106×ΔT×Lo1
Lo1=19×10611×106Lo2
[L2L1=Lo2Lo1 as difference in length is constant, given]
Lo1=1911Lo2 ... (3)
Also, Lo1Lo2=30 cm [Given]
1911Lo2Lo2=30 [from (3)]
811Lo2=30
Lo2=41.25 cm ... (4)
On putting (4) in (3),
Lo1=71.25 cm

Lengths of the rods at 0 C are 71.25 cm and 41.25 cm of steel and brass respectively.

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