Question

# A uniform metal rod is used as a bar pendulum. If the room temperature rises by ${10}^{0}\mathrm{C}$, and the coefficient of linear expansion of the metal of the rod is $2×{10}^{-6}\mathrm{per}\mathrm{degree}\mathrm{C}$, the period of the pendulum will have a percentage increase of

A

$-2×{10}^{-3}$

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B

$-1×{10}^{-3}$

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C

$2×{10}^{-3}$

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D

$1×{10}^{-3}$

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Solution

## The correct option is D $1×{10}^{-3}$Step 1: Given informationCoefficient of linear expansion, $\mathrm{\alpha }=2×{10}^{-6}\mathrm{per}\mathrm{degree}\mathrm{C}$Change in temperature, $\mathrm{\Delta T}=10°\mathrm{C}$Step 2: CalculationWe know,The fractional increment in time with temperature change:$\frac{∆\mathrm{t}}{\mathrm{t}}=\frac{1}{2}\mathrm{\alpha }∆\mathrm{T}$By substituting the values, we get $=\frac{1}{2}×2×{10}^{-6}×10\phantom{\rule{0ex}{0ex}}={10}^{-5}$The percentage change of period be:$%\left(\frac{∆\mathrm{t}}{\mathrm{t}}\right)=\left(\frac{∆\mathrm{t}}{\mathrm{t}}\right)×100\phantom{\rule{0ex}{0ex}}={10}^{-5}×100\phantom{\rule{0ex}{0ex}}={10}^{-3}\phantom{\rule{0ex}{0ex}}=1×{10}^{-3}$Therefore, option D is the correct choice.

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