Numerical and Graphical Analysis in Physical Science
In physics, we are often tasked with finding relationships
between measured quantities.
There are a number of ways to do this. Here we will look
at two – multipliers and graphing techniques.
Multipliers
When looking at numerical data tabulated sequentially, the
multiplier for value “A” is the number we must multiply the
first measured value by to get “A”.
Time (s)
Position (m)
1
4
2
2
8
3
3
12
4
4
16
Example
2
3
4
Here the
multipliers for
position are
equal to those
for time, so
Position α time
The statement “position α time” is called a
“proportionality statement” and is read “position is directly
proportional to time”. This means that as time changes,
position changes by the same factor (time doubles,
position doubles, etc.)
Example 2
Voltage (V)
Power (W)
1
3
2
2
12
4
3
3
27
9
4
4
48
16
Power multiplier = Voltage multiplier squared, so ...
Power α Voltage2
Or P α V2
How is this read?
“Power is directly proportional to the square of Voltage.”
Graphical Analysis
A more accurate method of finding this type of
relationship is to use graphing.
Steps:
1. Plot a scatter graph (that means just the data points
for now, no lines yet). Plot the dependent variable on the
y-axis unless instructed otherwise.
2. Draw either a straight line of best fit with a ruler, or a
curve of best fit freehand.
3. Compare your graph to one of the four common
graph types to make a reasonable guess at the
relationship between the variables.
Common Graph Types
Linear
Root
5
45
4.5
40
4
35
3.5
30
3
25
2.5
20
2
15
1.5
10
1
5
0.5
0
0
0
5
10
15
20
25
0
5
10
Inverse
15
20
25
Power
1.2
450
1
400
350
0.8
300
0.6
250
200
0.4
150
0.2
100
50
0
0
5
10
15
20
25
0
0
5
10
15
20
25
4. For a linear graph, simply determine the value of the
slope (called the “constant of proportionality”) and the
equation of the line now describes the accurate
relationship.
5. For the other three graph types, “Power” is of the
n
form y α xn (n>1), “Root” is of the form y
x
1
and “inverse” is of the form y
n
x
6. To find the value of n, we can now replot “y” vs. some
function of “x” that we choose based on the shape of the
first graph. For example, if the first graph looks like a
power curve we can plot y vs. x2.
7. Once we have a linear plot, we determine the slope
and write the equation of the line. For the power
example, if y vs. x2 gives us a straight line, the equation
would be y = kx2 + b, where k is slope and b is the y-int.
For the given data set, use graphing techniques to
determine the relationship between Force and
Frequency.
Force (N)
Frequency (Hz)
4
0.75
8
1.06
12
1.30
16
1.50
20
1.68
24
1.84
28
1.99
32
2.12
36
2.25
40
2.37
44
2.49
48
2.60
Graph 1 – Frequency vs Force
Frequency vs Force
for Mass Moving in a Circle
3.00
2.50
Frequency (Hz)
2.00
1.50
1.00
0.50
0.00
0
10
20
30
Force (N)
40
50
60
This graph is clearly not linear, so at least one more graph
is required. I'd say it looks like a root graph, so I'll plot a
new graph of Frequency vs Force (in math class you'd
call this a y = x graph).
Frequency vs Force^0.5
for Mass Moving in a Circle
3.00
2.50
Frequency (Hz)
2.00
1.50
1.00
0.50
0.00
1.00
2.00
3.00
4.00
Force^0.5 (N^0.5)
5.00
6.00
7.00
8.00
Now we have a linear graph, so we can calculate the
slope.
y 2− y 1
slope =
x 2− x 1
2.5− 0.5
slope =
6.7− 1.4
= 0.38
So the constant of proportionality is 0.38 Hz/N^0.5. This
gives us the equation that describes the relationship
between force and frequency:
f = 0.38
F
Or, taking the
reciprocal of 0.38,
F = 2.63 f