### Experimential Classificiation

## Reference: Conducting Proportionality Investigations

Scientific experiments are often conducted to reveal the relationship between two quantities. This is particularly useful when there is no pre-existing theory which describes a relationship. A successful proportionality investigation will allow the scientist to develop an accurate mathematical model (formula) for a physical process or system. This model can be tested multiple times until its validity is no longer questioned. At this point the formula or model is elevated to the status of scientific law. When enough scientific laws and conceptual understanding related to a given phenomenon are established, they are grouped together into what is called a scientific theory. Note that a theory is not theoretical anymore… it is actually a volume of well-established work. Within these experiments the scientist will choose a physical quantity to be the independent variable (IV). This is the quantity which is altered systematically from one trial run to another. A second quantity that will then hypothetically be affected by these changes is known as the ‘dependent variable’ (DV). The scientist’s experimental design must take into account a means of measuring both IV and DV precisely and accurately. Also, any other potential variables must remain constant… these are called experimental controls. To help guide the scientist in the investigation it can be useful to construct a scientific hypothesis. This is a general predictive statement about the relationship between IV and DV based on observation and reason. After data collection is complete, the relationship between the IV and DV is traditionally determined by constructing an x-y scatter plot. The IV is plotted on the x-axis; the DV is plotted on the y-axis. A regression curve is then fit to the plotted data. It is up to the scientist to determine what curve will best fit the data. Often a computer is used to do this most accurately. This year the vast majority of physical proportionalities under investigation may be fit with one of the following curves:Name of Proportionality | Shape of Curve | Algebraic Function | Graph |
---|---|---|---|

linear | straight line | $y = k x$ | |

square or (quadratic) | parabola | $y = k x^2$ | |

square - root | parabola | $y = k \sqrt{x}$ | |

inverse | hyperbola | $y = \frac{k}{x}$ | |

inverse square | hyperbola | $y = \frac{k}{x^2}$ | |

inverse square - root | hyperbola | $ y = \frac{k}{\sqrt{x}}$ |

After determining the proportionality displayed by the graph, a scientific formula may be expressed, with symbols substituted in for the mathematical placeholders: x, y, and the proportionality constant, k.

For example: When studying a set of circular shapes, a graph of circumference (C) vs.

diameter (D) would produce a linear relationship with a slope of 3.14.

This would be expressed as: $$ y = kx $$

$$C = 3.14 D$$

$$C = \pi D$$

## Reference: Using Linearization to Determine Proportionality Constants

On the previous page it was stated that many physical relationships can be expressed through simple algebraic formulas. Experimentation allows a scientist to determine what that formula is. The example on the previous page concerned the formula for the relationship between the circumference of a circle and the diameter. This formula is linear with the slope of the line being the well-known constant ‘pi’.When the data collected produces a relationship which is non-linear, a process of linearization must take place in order to determine the value of the proportionality constant.

For example: When studying a set of spherical shapes, a graph of surface area (A) vs. radius (D) would produce a non-linear, parabolic relationship.

This would be expressed as: $y = kx^2$

$A = kr^2$

Since the graph is parabolic we cannot equate the slope of the line to the value of ‘k’. Therefore the equation is incomplete at this point.

However, it can be noticed that: $k = \frac{A}{r^2}$ This manipulation of [$A = kr^2$ $\longrightarrow$ $k = \frac{A}{r^2}$] is referred to as linearization.

A graph of A vs. $r^2$ would look like this

The slope of this graph provides the value of k: $A = 12.57r^2$

And permits the scientist to find the final formula. $A = (4\pi)r^2$

## Reference: Measuring an Unknown Value via Experimentation

Many experiments are conducted to determine the value of certain quantities. These quantities may be typically values that are impossible to measure through direct ordinary means.For example:

mass of the Sun = 1 990 000 000 000 000 000 000 000 000 000 kg = 1.99 x10

^{30}kg

wavelength of light = 0.000 000 550 m = 5.50 x 10

^{-7}m

A simple scientific process allows a scientist to measure the unknown quantity indirectly by measuring two other quantities. These quantities are the IV and DV of the experiment and are known as the raw data. This raw data is manipulated to produce some derived data. When graphed… the derived data produces a linear relationship with a slope equal to the unknown measurement.

Say I wanted to measure the height of a flagpole – impossible to measure directly. I could set up a short stick a measureable distance away from the tree. This distance (a) could be varied as the independent variable. Together, the geometry of the pole and stick would form two similar right triangles. A line of sight connecting the top of the pole to the top of the stick would define the hypotenuse of each triangle. As the independent variable (a) changes, the measureable distance (c) will change as well. This could be assigned the independent variable status. Since the stick has a constant value (h) it is deemed to be an experimental control. Starting from the known relationship between legs of similar triangles:

$\frac{H}{b} = \frac{h}{c} \longleftarrow$ Orginal Formula

$H = \frac{hb}{c} \longleftarrow$ Manipulated Formula

$H = \frac{h( a + c)}{c} \longleftarrow$ Adjusted Formula it Include IV

$H = \frac{h( a + c)}{c} \longleftarrow$ Final formula to place variable in both numerator and demoninator

The right side of the final formula will then have a numerator (N) and a denominator (D). These quantities need to be calculated as the derived data. So our data table(s) should look something like this:

Control | Raw Data | Derived Data | ||
---|---|---|---|---|

Stick height h (ft) | a(ft) | c(ft) | h(a + c) (ft^{2}) |
c (ft) |

2 | 10 | 0.9 | 21.7 | 0.9 |

15 | 1.3 | 32.6 | 1.3 | |

20 | 1.7 | 43.5 | 1.7 |

When the derived data is graphed, with c on the x-axis and h(a+c) on the y-axis… the slope of the line will be the height of the flagpole. This is by design.

$H = \frac{h(a +c)}{C}$ looks just like ... $slope = \frac{h(a +c)}{C}$

## Reference: Producing Experimental Confirmation of Theory

After the discovery of physical proportionalities and development of scientific formula a scientist may wish to test the predictive power of the work or expand the work into new areas. To do so, the scientist will complete theoretical derivations through mathematical manipulation of one or more formula. These derivations will lead to new a formula that needs to be tested. The derived formula is the hypothesis in this scenario.The theoretical formula can be used to generate theoretical data by assigning a range of values to one variable and calculating another.

The scientist will then devise an experiment to test the theoretical data. Within such an experiment the scientist will once again assign one quantity to be the 'independent variable' (IV) and another to be the 'dependent variable' (DV). From the experiment, the scientist will gather experimental data.

Finally, the theoretical data and the experimental data are plotted simultaneously on a single x-y scatter plot. Typically, the theoretical data is presented with a smooth theoretical curve and the experimental data is presented as data points with associated error bars. The degree to which the experimental data points match up with the theoretical curve gives the scientist an indication whether the scientific work and derivations were correct.

For example, the following graph shows a theoretical curve based on the formula: $T = 2\sqrt{L}$

To generate this curve a range of pendulum lengths values between 10 cm and 100 cm were inserted into the formula to produce a corresponding range of T-values. The data points were then collected experimentally. Note that the data points don't fall exactly on the theoretical curve. Error bars are included on the data points… and with few exceptions, the theoretical line passes through all of them.