Consider a fully insulated (both laterally and on the ends) bar of length L, that has been heated to an initial temperature distribution of $-x(x-L)$. Find the total heat in the bar for all time.
Above is my problem statement. Here is what I have gathered:
- The bar is fully insulated, so not heat can enter nor escape the system, other than the initial temperature.
- The initial temperature distribution implies zero degrees on the ends of the bar and maximum temperature in the centre.
- Putting the two together, I assume that the bar will eventually come to a non-zero uniform, constant temperature.
So, am I correct to say that the total heat is simply the integral of $-x(x-L)$ for $0$ to $L$, or am I missing something?
HINT: Let $\rho$ be the linear mass density & $c$ be the specific heat of the bar then consider a bar element of length $dx$
The total heat in the bar $$=\int_{0}^{L} c\ dm T_x=\int_{0}^{L} c(\rho dx) T_x=\rho c\int_{0}^{L}T_x\ dx$$ Put the value of the temperature distribution function $T_x$