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 string'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?

Explanation

Comparing the given equation with equation of a travelling wave, `kg//m` The average power over a period is, `P = 1/2 (rho S) omega^2 A^2v`
Substituting the values, we have
`P = 1/2 (0.2) (4)^2 (8 xx 10^(-2))^2 (2.236) = 2.29 xx 10^(-2)J//s`
Now let it take t second to raise the temperature of 1 kg water by one degree kelvin. Then `Pt = ms Deltat`
Here, s = specific heat of water `= 4.2 xx 10^3 J//kg -K`
`:. t = (msDeltat)/(P xx 1/2) = ((1) (4.2 xx 10^3) (1))/(1.145 xx 10^(-2)) ~~ 4.2` day