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A 20 kg load is suspended from a copper ...

A 20 kg load is suspended from a copper wire of radius 1 min. If the wire suddenly snaps, does its temperature change? Calculate this change in temperature. For copper Young's modulus `=12xx10^(10)"N.m"^(-2),` density `="9000 kg.m"^(-3)`, specific heat = `"100 cal.kg"^(-1).K^(-1)`. Mechanical equivalent of heat = `"4.2 J.cal"^(-1),g=9.8m.s^(-2)`.

Text Solution

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The work done in order to increase the length of the wire by the application of force is stored as potential energy inside the wire. If the wire suddenly snaps, then the accumulated potential energy of the wire is converted into heat energy. Hence, the temperature of the wire increases.
Potential energy stored in the wire
= work done in stretching the wire
`=(1)/(2)xx" applied force"xx"elongation"=(1)/(2)xxFxxl`
We know that, `Y=(F.L)/(A.l)`
`"or, "l=("F.L")/("Y.A")=(20xx9.8xxL)/(12xx10^(10)xx3.14xx(10^(-3))^(2))`
Potential energy stored in the wire
`=(1)/(2)xx20xx9.8xx5.2xx10^(-4)xxL=5.1xx10^(-2)xxLJ`
Equivalent heat energy, `H=(5.1xx10^(-2)xxL)/(4.2)"cal"`
Let the increase in tempeature of the wire be `theta` K.
`therefore" "H=ms theta`
`=pixx(10^(-3))^(2)xxLxx9000xx100xx theta`
`"or, "(5.1xx10^(-2)xxL)/(4.2)=pixx(10^(-3))^(2)xx9000xxLxx100xxtheta`
`"or, "theta=(5.1xx10^(-2))/(4.2xxpixx(10^(-3))^(2)xx9000xx100)`
`=0.0043K`
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