A string having bath!

Classical Mechanics Level 3

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 = $\SI[per-mode=symbol]{4.2}{\kilo\joule/\kilo\gram\kelvin}$
Neglect the effect of gravity.

The answer is 3668500.

1 solution

Istiak Reza
Jun 24, 2016

We know that average power transmitted by a traveling wave per unit length is $P=\frac {E}{t}=\frac {1}{2}µw^2a^2v$
where $E$ is the transmitted energy, $\mu$ is the mass per unit length, $\omega$ is the angular frequency of the wave, $A$ is the wave amplitude, $v$ is the wave speed.
We can find the wave speed $v$ by the equation $v=\sqrt{F/µ}$ where $F$ is the tension of the string.

As the efficiency of heat transfer is $50\%$ , half of the total transmitted energy will be transferred to the water as heat.
Now we may write $\begin{aligned} \frac12 E&=Q \\ &=mc \Delta T \\ \frac12 Lµw^2a^2vt&=2mc\Delta T \\ t&=\frac{4mc\Delta T}{Lµw^2a^2v} \end{aligned}$

Plugging in the values we obtain $t\approx 36.685\times10^5$ .

A string having bath!

The answer is 3668500.

1 solution

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