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A thin rod of negligible mass and area o...

A thin rod of negligible mass and area of cross sectional `4 xx 10^(-6)m^(2)`, suspended vertically from one end has a length of 0.5 at `100^(@)C` the rod is cooled at `0^(@)C`, but prevented from contracting by attaching a mass at lower end. Find (i) mass (ii) the energy stored in rod. Given for rod, `y = 10^(11) Nm^(-2)`, coefficient of linear expansion `= 10^(-5) k^(-1)` & `g = 10 ms^(-2)`.

Text Solution

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(i) `A = 4 xx 10^(-6) m^(2) , l = 0.5 m`
`Delta T = 100 - 0 = 100^(@)C = 100 K`
`y = 10^(11) Nm^(-2), alpha = 10^(-5) K^(-1)`
Change in length,
`Delta l = l alpha Delta T = 0.5 xx 10^(5) xx 100`
`= 5 xx 10^(-4)m`
as `y = ("stress")/("strain")`
`"Stress" = (y Delta l)/(l) = y xx alpha Delta T`
`= 10^(11) xx 10^(-5) xx 100 = 10^(8) Nm^(-2)`
Stretching force,
`F = "stress" xx "area" = 10^(8) xx 4 xx 10^(-6)`
`= 4 xx 10^(-2) N`
but `F = Mg " " :. M = 4 xx 10^(-2)`
`M = (4 xx 10^(2))/(g) = (4 xx 10^(2))/(10) = 40 kg`
(ii) Energy stored in the rod
`= (1)/(2)F xx Delta l = (1)/(2) xx 4 xx 10^(2) xx 5 xx 10^(-4) = 0.1 J`
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