When the independent variable I and the dependent variable D are related by a power law, i.e. by:
D
= c I p
(An independent variable is a physical quantity that you can control, you can set its value yourself, like the distance you want an object to fall for example; a dependent variable is a physical quantity that varies as a result of varying the independent variable, but you can't control it directly, like the time it takes for the object to fall, for instance.)
1) Graph ln D vs ln I. Make sure that the origin appears on your graph
… and say so! Both the title of
the graph, and the labels on your axes, should say ln D rather than D (and the same
goes for I!).
This is because, taking the ln of
the power law:
ln
D = ln(c I p)
gives: ln D = ln c + p ln I
ln D is a variable
ln c is a constant
p is a constant
ln I is a variable
So this equation is of the form:
variable
= constant + constant (different
variable)
And THAT is the definition of the
equation of a straight line. In high school, you were probably asked to call
the first variable y, and taught that the first constant is the y-intercept, and
the second one the slope, like so:
y
= y-intercept + slope . x
In this course don't use
"m" for the slope: we'll use it all the time to mean mass.
So what does this mean? It means
that if you take y = ln D and x = ln I, your data points will form a straight
line (more or less).
2) Draw the best fit line (or "line of best fit")… using a
ruler ;-)
3) From the graph itself, find:
-
the slope of the best fit line.
- use points on the best fit line
that are NOT data points.
- on your graph, encircle the
points you use to find the slope.
- show your calculation of the
slope explicitly and in all the gory details, in your lab report (under
"calculations").
-
the y-intercept
- read it off the graph.
- encircle it on your graph.
watch out: make sure that what
you are reading off does correspond to x = 0!
4) Find the value of c and p from the y-intercept and the slope:
DO NOT SUBSTITUTE ANY EXPERIMENTAL DATA IN ANY EQUATION!!!
When students get a bad grade on their lab reports (D or worse), almost always it is because they did that.
When students get a bad grade on their lab reports (D or worse), almost always it is because they did that.
Instead, compare:
ln
D = ln c +
p . ln I
and: y =
y-intercept + slope . x
where I have indicated the
variables in red and the constants in green.
So provided that you plot ln D vs
ln I:
-
what is the y-intercept, in terms of c or/and p?
-
what is the slope, in terms of c or/and p?
Answering these questions
correctly allows you to write two equations:
y-intercept
= …
slope
= …
where the ... on the right hand sides should be expressions that contain p, c, or both.
where the ... on the right hand sides should be expressions that contain p, c, or both.
Solve them for c and p, that is
write c and p in terms of the slope or/and the y-intercept (hint: since the equation for a straight line is x = slope . x + y-intercept, the slope is the constant that multiplies what is on graphed on your x axis ):
c
=…
p
= …
Now in these equations, substitute the
values (i.e. numbers, like say 3.4 or 29.7) for the slope and the y-intercept of your best-fit line, i.e. those that
you found from your graph.
Give your result, that is, the equation:
with the correct values for c and p substituted in, with p rounded of to the nearest simple ratio (ex: if it's say 0.192, round off to 2; if it's 0.45, round off to 1/2). Also:
IMPORTANT FOR YOUR GRADE:
You MUST give in your lab report, under the section "calculations", ALL the equations that I have indicated above in light blue (correctly completed of course!).
This is what SHOW YOUR WORK means in this case.
If you do not do this, even if you get the right answer somehow, I will be forced to assume that you just got the lucky (or worse, got it from someone else!). It is your responsibility to prove that you understand and know how to apply the methods that you are expected to learn. Here, the way to do this is by doing and showing the calculations I indicated in light blue, and clearly give all of the following, in the calculations section:
- the value of the slope: (that would be a decimal number)
- the value of the y-intercept: (that would be a decimal number)
- the expression for the slope in terms of c or/and p (i.e. the coefficient and the power); that is you should actually write "slope = " and have an expression on the right hand side with c or/and p in it.
- the expression for the y-intercept in terms of c or/and p (y-intercept = … expression with c or/and p in it.
- solve the above two equations for p and c, in terms of the slope / the y-intercept.
- indicate the value of p and c, as decimal numbers.
- give the equation between the variables that you measured, that is something of the form D = c I superscript p (I can't superscript in this email sorry!) (round off p). Use the symbols for the variables, not D and I as I used on the blog.
c
=…
p
= …
Give your result, that is, the equation:
D
= c I p
with the correct values for c and p substituted in, with p rounded of to the nearest simple ratio (ex: if it's say 0.192, round off to 2; if it's 0.45, round off to 1/2). Also:
IMPORTANT FOR YOUR GRADE:
You MUST give in your lab report, under the section "calculations", ALL the equations that I have indicated above in light blue (correctly completed of course!).
This is what SHOW YOUR WORK means in this case.
If you do not do this, even if you get the right answer somehow, I will be forced to assume that you just got the lucky (or worse, got it from someone else!). It is your responsibility to prove that you understand and know how to apply the methods that you are expected to learn. Here, the way to do this is by doing and showing the calculations I indicated in light blue, and clearly give all of the following, in the calculations section:
- the value of the slope: (that would be a decimal number)
- the value of the y-intercept: (that would be a decimal number)
- the expression for the slope in terms of c or/and p (i.e. the coefficient and the power); that is you should actually write "slope = " and have an expression on the right hand side with c or/and p in it.
- the expression for the y-intercept in terms of c or/and p (y-intercept = … expression with c or/and p in it.
- solve the above two equations for p and c, in terms of the slope / the y-intercept.
- indicate the value of p and c, as decimal numbers.
- give the equation between the variables that you measured, that is something of the form D = c I superscript p (I can't superscript in this email sorry!) (round off p). Use the symbols for the variables, not D and I as I used on the blog.
PRACTICE EXAMPLES:
1) You need to find the mathematical relationship between two physical quantities, V and W. You can directly vary W, and V then varies as a result.
1) You need to find the mathematical relationship between two physical quantities, V and W. You can directly vary W, and V then varies as a result.
Write down on a piece of paper what the following must be, for you to be able to find their relationship:
- what you need to graph on your y axis: ln V (not just V!)
- what you need to graph on your y axis: ln V (not just V!)
- what you need to graph on your x axis: ln W (not just W!)
Now highlight with your mouse the lines above to check your answer.
Now let's assume you have graphed this and done so correctly. You have also found correctly that the best fit line of your graph has:
- a slope of 4.2
- a y-intercept at 1.7
Write down on a piece of paper what the mathematical relationship between V and W is.
V = 5.5 W 4
Now highlight with your mouse the line above to check your answer.
2) You need to find the mathematical relationship between two physical quantities, D and L. D varies when you vary L.
3) You need to find the mathematical relationship between two physical quantities, A and B. As you vary A, B changes as a result..
2) You need to find the mathematical relationship between two physical quantities, D and L. D varies when you vary L.
Write down on a piece of paper what the following must be, for you to be able to find their relationship:
- what you need to graph on your y axis: ln D
- what you need to graph on your y axis: ln D
- what you need to graph on your x axis: ln L
Now highlight with your mouse the lines above to check your answer.
Now
let's assume you have graphed this and done so correctly. You have also
found correctly that the best fit line of your graph has:
- a slope of 0.65
- a y-intercept at 7.5
Write down on a piece of paper what the mathematical relationship between L and D is.
D = 1800 L 2/3
Now highlight with your mouse the line above to check your answer.
3) You need to find the mathematical relationship between two physical quantities, A and B. As you vary A, B changes as a result..
Write down on a piece of paper what the following must be, for you to be able to find their relationship:
- what you need to graph on your y axis: ln B
- what you need to graph on your y axis: ln B
- what you need to graph on your x axis: ln A
Now highlight with your mouse the lines above to check your answer.
Now
let's assume you have graphed this and done so correctly. You have also
found correctly that the best fit line of your graph has:
- a slope of -0.9
- a y-intercept at 3.2
Write down on a piece of paper what the mathematical relationship between A and B is.
B = 24 A -1
Now highlight with your mouse the line above to check your answer.
