HOW TO FIND (EXPERIMENTALLY) THE VALUE OF A CONSTANT PHYSICAL QUANTITY, WHEN YOU ARE GIVEN AN EQUATION THAT CONTAINS IT.
In order to do this you need to draw a graph using data collected in a lab. However not any graph will do: you need to make sure that what you graph will lead to data points that form a straight line.
So you have to use a "trick".
Imagine that you were asked to determine experimentally a constant called m, and told that it obeys this awful looking equation:
I7 = 3 m a D5
What you need to do is:
1) First, identify what in this equation are variables, and what are constants.
About the variables: Identify which symbols represent the physical quantities that you can and want to make vary in the lab. There should be two of those.
Then identify which one of the two you will be controlling yourself: that's your independent variable.
The other one, whose value will change automatically when you vary the independent variable, is the dependent variable.
About the constants: Constants can be just numbers, or they can be things like π;, or physical constants like g.
OR they can also be things that could in principle vary. You have to watch out for those for two reasons:
- you will have to make sure that in fact do not, when you do the experiment. You need to keep them constant.
- you need to record their (constant) value.
In the example equation above for instance, I stands for the independent variable, D for the dependent one, m could in principle vary but we don't want it to, because we've decided we'd rather vary I instead for practical reasons. Of course in real life the variables are not conveniently called I and D for you. You have to figure out what plays the part of I and D in the equation given to you, from your knowledge of the experimental set-up.
For the sake of clarity, let me color coding the variables as red in the equation:
I7 = 3 m a D5
Then what we call the functions of these variables are the expressions involving them indicated in orange below:
I7 = 3 m a D5
The rest in white is an expression made up purely of constants, so it is something constant.
2) The first thing you need to do is to consider the expression that contains your dependent variable, and solve for it. Here it is D5 .(If you're lucky the equation may already be in that form, but that's not always true of course). In our example this results in:
D5 = 1⁄(3 m a) I7
3) Now comes the crucial step: decide what you need to graph on the y-axis and on the x-axis of your graph, in order to get a straight line.
Clearly, graphing D vs I won't work.
So you may be tempted to try and "get rid" of what seems to be causing a problem by, say, taking the power 1/5, or 1/7, to "get rid of" the powers of 5 and 7. The equation will look different, sure. But in fact that doesn't help either, because the shape of a graph (like it being straight), depends on the relationship between what you graph on the x axis and what you graph on the y-axis. And putting the equation in a different form doesn't change the relationship between the variables (D and I here).
So in order to know what you need to graph, you must understand what truly defines a straight line. Graphing one thing vs another will give you a straight line only if those two things are "linearly related" to each other, or "proportional" to one another - the two phrases mean the same thing.
"proportional" means: to differ from one another by only a constant factor; that the one is equal to the other, just multiplied by something constant. That is, we say that:
Clearly, graphing D vs I won't work.
So you may be tempted to try and "get rid" of what seems to be causing a problem by, say, taking the power 1/5, or 1/7, to "get rid of" the powers of 5 and 7. The equation will look different, sure. But in fact that doesn't help either, because the shape of a graph (like it being straight), depends on the relationship between what you graph on the x axis and what you graph on the y-axis. And putting the equation in a different form doesn't change the relationship between the variables (D and I here).
So in order to know what you need to graph, you must understand what truly defines a straight line. Graphing one thing vs another will give you a straight line only if those two things are "linearly related" to each other, or "proportional" to one another - the two phrases mean the same thing.
"proportional" means: to differ from one another by only a constant factor; that the one is equal to the other, just multiplied by something constant. That is, we say that:
expression A is proportional to expression B
when
expression A = something constant x expression B
And when that's the case, graphing expression A vs expression B will give you a straight line, with as its slope the value of that constant something.
That's what we mean by the equation of a straight line is:
That's what we mean by the equation of a straight line is:
Y = slope x X
But Y and X don't have to be something simple, they can be complicated expressions; only it is these complicated expressions that must be graphed then. Also, the slope must be constant!
Now that we know this, let's go back to our example (with variable and constant expressions).
Now that we know this, let's go back to our example (with variable and constant expressions).
D5 = 1⁄(3 m a) I7
Comparing it to the equation of a straight line, we see that what corresponds to:
is:
Y = slope x X
D5 = 1⁄(3 m a) I7
So IF you graph D5 on the y-axis, and I7 on the x axis, then (and only then!):
slope = 1⁄(3 m a)
Where "slope" stands for the slope of the line of best fit to the data points.
At this point, all you have to do to find m, is solve this last equation! You get:
m = 1/(3 a slope)
Then to find the value of m, which remember is what you were asked to find, you need to substitute the value of a (which hopefully you will have recorded in lab!) and the value of the slope (which on a graph by hand you need to find by circling 2 points on best-fit line and using them to calculate rise vs run; on a graph done in Excel is given by excel in the equation of the best fit line).
To sum up, here is what YOU would need to do BEFORE YOU START to write the lab report asking for you to find m, on a separate sheet of paper for yourself:
To sum up, here is what YOU would need to do BEFORE YOU START to write the lab report asking for you to find m, on a separate sheet of paper for yourself:
OTHER EXAMPLE:
In the example above the relationship between the variables was a power law. But this method works whatever the relationship is, as the next example shows:
You are told that:
er = (7 π)⁄c . V . ln (H)
You are asked to find c.
1) From knowing what all these symbols actually mean physically in the context that they are discussed, you can tell that r and H are the only physical quantities that can be varied, and from the set-up you can tell that H is the independent variable, and r the dependent one i.e. you can vary H yourself and r will vary as a result.
2) You solve for the expression that involves the dependent variable. That is er. But, it is already solved for, so in this case you don't have to do anything.
3) You compare this to the form of the equation for a straight line:
er = (7 π)⁄c . V . ln (H)
Y = slope x X
and conclude that you need to graph er on the y-axis, ln (H) on the x axis, and that in this case:
slope of the best fit line = 7 π V/c
Therefore you will find the value of c with:
c = 7 π V/slope
and substituting the value you found for the slope of the best fit line.
To sum up, here is what YOU would need to do BEFORE YOU START to write the lab report asking for you to find m, on a separate sheet of paper for yourself:
PRACTICE EXAMPLES:
1) You are told to measure the value of some constant quantity named k and you know that:
z = (3/2) k m5
where z and m are variables that you can measure. In the experiment, you can control the value of z, and m varies as a result. Write down what the following must be:
- what you need to graph on your y axis: m5
m is your dependent variable, so solve the equation for the expression that contains it: m5 = (2/(3 k)) z
That way the dependent variable ends up on the left hand side of your equation - ready to appear on the y-axis rather than the x-axis, as tradition requires.
If you decided to graph m on the y-axis, rather than m5 , this is wrong because m is not proportional to z: m is not equal to z times just a constant; it is m5 that is equal to z times a constant (where that constant is 2/(3 k)). So it is m5 that is proportional to z, hence graphing m5 vs z must yield a straight line.
- what you need to graph on your x axis: z
- the expression for the slope you will get IF you do graph this: slope =2/(3 k)
Remember the structure of a straight line: y = slope . x + y-intercept. So the slope is the constant expression that multiplies x, where by x we mean whatever is graphed on the x axis.
- the expression for k in terms of the slope: k = 2/(3 slope)
Now highlight with your mouse the lines above to check your answer.
2) You are told to measure the value of some constant quantity named c and you know that:
A3 = 5 c d
where A and d are variables that you can measure. In the experiment, you can control the value of d, and A varies as a result. Write down:
- what you need to graph on your y axis: A3
- what you need to graph on your x axis: d
If you decided to graph A on the x-axis, this is wrong because A is not proportional to d: A is not equal to d, just multiplied by a constant; it is A3 that is equal to d times a constant (where that constant is 5 c). So it is A3 that is proportional to d, hence graphing A3 vs d must yield a straight line.
- the expression for the slope you will get IF you do graph this: slope = 5c
Remember the structure of a straight line: y = slope . x + y-intercept. So the slope is the constant expression that multiplies x, where by x we mean whatever is graphed on the x axis.
- the expression for c in terms of the slope: c = slope / 5
Now highlight with your mouse the lines above to check your answer.
3) You are told to measure the value of some constant quantity named v and you know that:
b3 = v2 j5
where b and j are variables that you can measure. In the experiment, you can control the value of j, and b varies as a result. Write down:
- what you need to graph on your y axis: b3
- what you need to graph on your x axis: j5
Indeed b3 and j5 are proportional to one another since the one is equal to the other times a constant. So this graph must give a straight line.
- the expression for the slope you will get IF you do graph this: slope = v2
- the expression for v in terms of the slope: v = square root of the slope
Now highlight with your mouse the lines above to check your answer.
