HOW TO FIND THE EQUATION THAT RELATES TWO PHYSICAL VARIABLES
(after you have collected data in the lab)
Example: You want to know the
relation between the time t an object takes to fall and the height h it falls.
We say you want to know the time the object takes to fall as a function of that
distance.
So you have dropped the object
from a certain height, say 2 meters, several times, say 5 times. Each time
you've dropped it you have measured how long it took to fall. Then you have
averaged those times.
Then you've repeated this process
for several different heights.
So you now have a table with two
columns: one for the heights, one for the corresponding (averaged) times.
Independent and
dependent variables:
I will call these variables I and
D, for "independent variable" and "dependent variable",
respectively.
The independent variable is the one that you control directly,
the dependent variable the one that
changes as a result.
Example: If you measure how long
it takes an object to fall a certain distance:
-
the distance you drop the object from is the independent variable (you
choose where to drop the object from).
- how long it takes the object to
fall that distance is the dependent variable (it depends on the distance it
falls… but not how long you want it to take!).
TO FIND THE MATHEMATICAL RELATION BETWEEN I AND D:
This semester, assume that this
equation is a power law, that is, assume that:
D
= c I p
where "c" stands for
"coefficient", and "p" stands for "power".
Note that this is a power law,
NOT an exponential law. An exponential law is something like: D = eI,
and in a case like that D would vary incredibly fast as you vary I!
D and I are variables (otherwise
wanting to find a relation between them would be meaningless).
c and p are constants.
Finding the relation between D
and I now means finding the value of c and p.
Note: here I am going to keep
calling them D and I, but you must substitute (i.e. "plug in") the
symbol for yours straight away; in the example of the falling object, this
would give you:
h = c t
p
To find c and p, you must PLOT A GRAPH of the data. This graph
should be linear, a straight line: that is, YOU must CHOOSE what to
plot on the x axis and what to plot on the y axis, in such a way that the
"data points" are going to more or less form a straight line.
This implies that (usually) you
must NOT graph D vs I. This is because this will not give you a straight line,
unless p happens to be 1 and you "know" that (it is told to you in the
lab manual).
No matter what, NEVER substitute
(i.e. "plug-in") DATA into ANY equation. If you do, even if you
find the right answer, you will get exactly as many points as if you had
done nothing at all: 0!
When students get a D or worse on their lab report, 80% of the time it is
because they did that.
What graph you need to plot, that
is what you need to graph on the x axis and on the y axis, depends. There are
two situations, hence two methods.
SITUATION 2: You don't know the coefficient c, but you do know the power p.
You are
asked to find experimentally the value of some constant quantity.
Link
