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A thin string is held at one end and osc...

A thin string is held at one end and oscillates so that,
`y(x = 0, t) = 8 sin 4t(cm)`
Neglect the gravitattional force. The dtring's linear mass density is `0 .2 kg// m` and its tension is ` 1 N`. The string passes through a bath filled with `1 kg` water. Due to friction heat is transferred to the bath. The heat transfer efficiency is `50%`. Calculate how much time passes before the temperature of the bath rises one degree kelvin?

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The correct Answer is:
A, B, C, D
Comparing the given equation with equation of a travelling wave,
`y= A sin (kx +- omegat)` at `x = 0` we find,
`A = 8 cm =8xx 10^(-2) m`
`omega = 4 rad//s`
Speed of travelling wave, `v = sqrt((T)/(mu)) =sqrt((1)/(0.2))=2.236 m//s`
Further, `rhoS = mu = 0.2 kg//m`
The average power over a period is
`P = (1)/(2)(rhoS) omega^(2) A^(2)v`
Substituting the values, we have
`p=(1)/(2)(0.2)(4)^(2) (8xx10^(-2))^(2) (2.236)`
`=2.29 xx 10^(-2)J//s`
The power transferred to the bath is,
`P'= 0.5P =1.145 xx 10^(-2) J//s`
Now let, it takes t second to raise the temperature of `1 kg`water by `1` degree kelvin`. Then
Here, `s = specified heat of water `= 4.2 xx 10^(3) J//kg-^(@) k`
`t=(msDeltat)/(P') = ((1)(4.2 xx 10^(3)(1))/(1.145 xx 10^(-2))`
`=3.6 xx 10^(5) s~~4.2 day`
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