A thin rod of negligible mass and area of cross-section 4x10-6m2, suspended vertically from one end, has a length of 0.5m at 100 0C. The rod is cooled to 00C but prevented from contracting by attaching a mass at the lower end. The value of this mass (in kg) is( Given, coefficient of linear expansion is 10-5 0C-1, Young’s modulus is Y = 1011 Nm-2 and g=10ms-2)

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A thin rod of negligible mass and area of cross-section 4x10^-6m², suspended vertically from one end, has a length of 0.5m at 100 ⁰C. The rod is cooled to 0⁰C but prevented from contracting by attaching a mass at the lower end. The value of this mass (in kg) is( Given, coefficient of linear expansion is 10^-5 ⁰C^-1, Young’s modulus is Y = 10¹¹ Nm^-2 and g=10ms^-2)

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answer is 40.

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Detailed Solution

The change in length due to decrease in temperature,
$Δ l_{1} = L α Δ θ = (0.5) (10^{- 5}) (0 - 100)$

$Δ l_{1} = - 0.5 \times 10^{- 3} m (i)$
Negative sign implies that length is decreasing. Now let M be the mass attached to the lower end.

$\frac{(\frac{M g}{A})}{\frac{∆ l}{l}} = 10^{11}$

$\Rightarrow M g = \frac{0.5 \times 10^{- 3}}{0.5} \times 10^{11} \times 4 \times 10^{- 6} = 400$

$M = 40 kg$

Then, change in length due to suspension of load is 40kg

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