Formulae from Experimental Data

When scientists & engineers try to find relationships between variables in experimental data the figures are often very large or very small and drawing meaningful graphs can be difficult.

The graphs often take exponential form so this adds to the difficulty.By plotting log values instead we often convert from

to

Example The variables Q and T are known to be related by a formula in the form

T = aQn

The following data is obtained from experimenting

Q 5 10 15 20 25

T 300 5000 25300 80000 195300

Plotting a meaningful graph is too difficult so taking log values instead we get ….

log10Q 0.7 1 1.2 1.3 1.4

log10T 2.5 3.7 4.4 4.9 5.3

This gives us the following graph from which we can get a line of best fit

log10Q

log10T

m = 5.3 - 2.5 1.4 - 0.7

= 2.8/0.7 = 4 (a,b) = (1,3.7)

Since the graph does not cut the y-axis use Y – b = m(X – a)

where X = log10Q and Y = log10T ,

Y – b = m(X – a)

log10T – 3.7 = 4(log10Q – 1)

log10T – 3.7 = 4log10Q – 4

log10T = 4log10Q – 0.3

log10T = log10Q4 – log10100.3 law3

log10T = log10Q4 – log102

log10T = log10(Q4/2) law2

T = 1/2Q4