Calculator Question: Graphing an Equation for Heat Conduction in a Rod

In summary: This will give an approximation, but it may be good enough for your purposes.In summary, the problem involves plotting u vs. x and u vs. t for a rod cooling off, and determining the amount of time it takes for the rod to reach no more than 1 degree C. The equation given is used, but the calculator skills are not sufficient to graph the equation and it may be necessary to use a computer program like Maple instead.
  • #1
bossman27
205
0

Homework Statement



For the rod in Problem 10 (already solved this, see below):
(a) plot u vs. x for t= 5, 10, 20, 40, 100, and 200
(b) plot u vs. t for x= 10, 20, and 30
(c) how long does it take for the entire rod to cool off to a temp. of no more than 1 degree C?


Homework Equations


u(x,t) = ([itex]\frac{160}{∏^{2}}[/itex])[itex]\sum(\frac{1}{n^{2}})sin(\frac{n∏}{2})sin(\frac{n∏x}{40})e^{(\frac{-n^{2}∏^{2}}{1600}t)}[/itex]



The Attempt at a Solution



I apologize in advance for my barbaric calculator skills.

I plugged the equation above into my TI-89, u(x,t) = y1, as written (including the limits n=1 to ∞). To try to do (a), I simply substituted the numerical values in for the t variable, but receive the "undefined variable" error message when I attempt to graph it. Although I don't have much experience using graphs for series equations, it seems that since the only variable in the y1 equation is x, and the function is supposed to be defined in terms of x, I don't know what the fix is. Would I just be better off going over to the computer lab and trying doing these problems on Maple or something?
 
Physics news on Phys.org
  • #2
bossman27 said:

Homework Statement



For the rod in Problem 10 (already solved this, see below):
(a) plot u vs. x for t= 5, 10, 20, 40, 100, and 200
(b) plot u vs. t for x= 10, 20, and 30
(c) how long does it take for the entire rod to cool off to a temp. of no more than 1 degree C?


Homework Equations


u(x,t) = ([itex]\frac{160}{∏^{2}}[/itex])[itex]\sum(\frac{1}{n^{2}})sin(\frac{n∏}{2})sin(\frac{n∏x}{40})e^{(\frac{-n^{2}∏^{2}}{1600}t)}[/itex]



The Attempt at a Solution



I apologize in advance for my barbaric calculator skills.

I plugged the equation above into my TI-89, u(x,t) = y1, as written (including the limits n=1 to ∞). To try to do (a), I simply substituted the numerical values in for the t variable, but receive the "undefined variable" error message when I attempt to graph it. Although I don't have much experience using graphs for series equations, it seems that since the only variable in the y1 equation is x, and the function is supposed to be defined in terms of x, I don't know what the fix is. Would I just be better off going over to the computer lab and trying doing these problems on Maple or something?

Yes. And you may have to settle for a large upper limit for the sum instead of using ##\infty##.
 

1. How do I graph an equation for heat conduction in a rod?

To graph an equation for heat conduction in a rod, you will need to use a scientific calculator that has graphing capabilities. Input the equation into the calculator and specify the variables that represent the temperature, length, and time. Then, use the graphing function to plot the equation on a graph.

2. What is the equation for heat conduction in a rod?

The equation for heat conduction in a rod is Q = (kAΔT)/L, where Q is the amount of heat transferred, k is the thermal conductivity of the material, A is the cross-sectional area of the rod, ΔT is the difference in temperature between the ends of the rod, and L is the length of the rod.

3. How do I determine the thermal conductivity of a material for the heat conduction equation?

The thermal conductivity of a material can be determined through experimentation or by looking it up in a materials database. It is a measure of how well a material can conduct heat and is typically measured in units of watts per meter-kelvin (W/mK).

4. Can I use the same equation to graph heat conduction in different materials?

Yes, the heat conduction equation can be used for any material as long as the thermal conductivity and dimensions of the material are known. However, keep in mind that different materials may have different thermal conductivities, which will result in different rates of heat conduction.

5. How can I use the graph to analyze heat conduction in a rod?

The graph can help you visualize the relationship between heat transfer, time, and temperature for a specific material and rod length. You can use it to compare different materials or rod lengths and see how they affect the rate of heat conduction. It can also be used to predict the temperature at a specific time or length along the rod.

Similar threads

  • Calculus and Beyond Homework Help
Replies
1
Views
1K
  • Calculus and Beyond Homework Help
Replies
5
Views
168
  • Calculus and Beyond Homework Help
Replies
2
Views
674
Replies
4
Views
594
  • Calculus and Beyond Homework Help
Replies
3
Views
851
  • Calculus and Beyond Homework Help
Replies
2
Views
836
  • Calculus and Beyond Homework Help
Replies
1
Views
646
  • Calculus and Beyond Homework Help
Replies
9
Views
1K
  • Calculus and Beyond Homework Help
Replies
3
Views
495
  • Calculus and Beyond Homework Help
Replies
1
Views
1K
Back
Top