# A string having bath!

A thin string is held at one end and oscillates vertically when driven by a motor so that $y(x=0,t)=8\sin4t\si{\ \centi\meter}$ The string's linear mass density is $$0.2\text{ kg m}^{-1}$$, its tension is $$1\text{ N}$$, and its length is $$\si{10\ \centi\meter}$$.

Suppose the string is now driven by the same motor inside a bath filled with $$\si{1\ \kilo\gram}$$ water. Due to friction heat is transferred to the bath with a heat transfer efficiency of 50%. Calculate how much time (in seconds) passes before the temperature of the bath rises by $$\si{1\ \celsius}$$.

Details and Assumptions:

• Specific heat of water = $$\si{4.2\ \kilo\joule/\kilo\gram\per\kelvin}$$
• Neglect the effect of gravity.
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