Question

In a particular set of experimental trials, students examine a system described by the equation given below. For experimental control, in these trials all quantities except d and Δt are constant.

If d is made three times larger, does the equation predict that Δt will get larger or smaller? By what factor will Δt change, if at all?

What pattern of proportionality of Δt to d does the equation predict?

To display this proportionality as a straight line on a graph, what quantities should you plot on the horizontal and vertical axes?

What expression represents the theoretical slope of this graph? (Use k, L, Q, Th, and Tc as necessary.)

Answer 1

Equation:

Q kπ (d^2)(Th - Tc)
----- = -------------------------
Δt 4L


Given that all quantitiies except d and Δt are constant you can write:

1
------- = C (d^2) => Δt = D / (d^2) (where D = 1/C)
Δt


If d is made three times larger, does the equation predict that Δt will get larger or smaller? By what factor will Δt change, if at all?


Δt = D / (d^2)


Now d' = 3d => Δt' = D / (3d)^2 = D / [9 d^2] = D/(d^2) * 1/9 = Δt / 9


Δt' = Δt / 9 => It predicst that Δt will become smaller and the factor of change is 1/9



What pattern of proportionality of Δt to d does the equation predict


The equation predicts that Δt is inversely related to d^2.



To display this proportionality as a straight line on a graph, what quantities should you plot on the horizontal and vertical axes?


You should plot Δt on the vertical axis and 1/(d^2) on the horizontal axis.



What expression represents the theoretical slope of this graph? (Use k, L, Q, Th, and Tc as necessary.)


The theoretical slope of that graph is what I called D, which I am going to calculate now step by step in terms of k, L, Q, Th and Tc:



Q kπ (d^2)(Th - Tc)
----- = ------------------------- =>
Δt 4L


4QL 1
Δt = -------------- * ----
kπ(Th-Tc) d^2


=> the slope is 4QL / [kπ (Th - Tc) ]