Linearization of Non Linear Equation

[BlackJackal]

Ok so here is the problem:

You have six steel bearing varying in size and you have their mass and diameter. When you graph the data you see that the mass (in Grams) goes up exponentially as diameter (in cm) goes up. Below is the table of the Mass and Diameters

M=.44 D=.4
M=2.04 D=.8
M=8.35 D=1.2
M=21.73 D=1.7
M=28.35 D=1.9
M=80.06 D=2.7

Ok so now you are supposed to graph the data above, which I did, and then you need to linearize the data, describe the importance of the slope of the linearized graph, and express the liearization as a new function.

Before we go any further I know that what the linearized data is supposed to be, because the proffesor told me. However I don't understand how he got it or what it means. Below is a list of points on the graph of the linearized function. BTW the second value is just M but I have no idea why that is.

D=2.7 LD=4.31
D=1.9 LD=3.05
D=1.7 LD=2.79
D=1.2 LD=2.03
D=.8 LD=1.27
D=.4 LD=.76

When you graph this you will see that the data is in the form of a line with slope of approx. 1.5. I have no idea the principles behind these operations and need help.

Thanks in advance.

 

[StatusX]

It doesn't seem to fit the second set of numbers you gave, but to linearize a set of exponentially increasing numbers, you plot their logs. For example, if your numbers were (1,10), (2,100), (3,1000),..., this is an exponential curve, but plotting the log base ten of the y components, (1,1),(2,2),(3,3),... , gives a linear curve (aka line).

 

[Astronuc]

I presume you are wanting to use a semi-log plot as StatusX indicated.

Take the log (base 10) or ln (natural log) of either dependent or independent variable or both if necessary, and see if you get a straight line.

The mass, M, of a sphere of a material is simply the density \rho * volume, where volume is \frac{\pi}{6}D^3.

One can take Log(M) which takes the form a + b log D, which yields a straight line.