Total thermal energy from heat equation

In summary, the conversation discusses deriving E(t) using the heat equation and integrating the right-hand side of the original equation. The sin(5t) term is mentioned, and a hint is given to find the partial derivative of E(t) with respect to t and integrate to find E(t), which requires determining a constant of integration. The conversation also mentions solving for E(0) using the given equation.
  • #1
Conservation
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Homework Statement


upload_2016-9-28_19-53-17.png


Homework Equations


Heat equation

The Attempt at a Solution


I can derive E(t) to get integral of du/dt over 0 to L, which is the same as integrating the right hand side of the original equation (d2u/dx2+sin(5t); while this allows me to take care of the d2u/dx2, I don't know what to do with the sin(5t) term or proceed from there.

Thanks.
 
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  • #2
Hi, mmm ##\int_{0}^{L}\sin{5t}dx=L\sin{5t}## ...
 
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  • #3
Another hint, when you find ##\frac{\partial}{\partial t} E(t)## as a function of ##t## you must integrate in order to find ##E(t)##, this gives you the problem to determine a constant of integration ... you can solve this calculating ##E(0)=\int_{0}^{L}u(x,0)dx=\int_{0}^{L}1+100\sin{\left(\frac{2\pi x}{L}\right)}dx## ...

Ssnow
 
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Related to Total thermal energy from heat equation

What is total thermal energy?

Total thermal energy is the sum of all the internal kinetic and potential energies of the molecules in a substance. It is a measure of the overall heat energy present in a system.

How is total thermal energy related to the heat equation?

The heat equation is a mathematical representation of how heat energy is transferred in a system. It takes into account factors such as temperature, heat flux, and thermal conductivity. By solving the heat equation, the total thermal energy in a system can be calculated.

What are the units of total thermal energy?

The units of total thermal energy depend on the system being studied. In the SI system, it is measured in joules (J). In the imperial system, it is measured in British thermal units (BTU).

How does the total thermal energy change in a system?

The total thermal energy in a system can change through heat transfer processes such as conduction, convection, and radiation. It can also change through chemical reactions, phase changes, and work done by external forces. The change in total thermal energy is quantified by the first law of thermodynamics.

How is total thermal energy used in practical applications?

Total thermal energy plays a crucial role in numerous practical applications, such as calculating the efficiency of power plants, designing heating and cooling systems, and understanding weather patterns. It is also used in the development of insulation materials and energy-efficient buildings.

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