AP Physics 1: Introduction 7: Basic Graphing
Summary
TLDRThis video discusses the importance of graphing data, focusing on why and how we graph to understand relationships between variables. Using an example from a classroom lab, it walks through graphing diameter vs. circumference of round objects, explaining key graphing principles like independent and dependent variables, scaling axes, scatter plots, and finding the line of best fit. The presenter also demonstrates how to calculate slope and derive a mathematical model, emphasizing the connection between the graph and the well-known formula for circumference, C = πD.
Takeaways
- 📊 Graphing data helps identify the relationship between two quantities.
- 📚 In high school, graphing was often done for grades, but the real purpose is understanding relationships in data.
- 🧮 In physics, the goal of graphing is to see how two quantities are related, such as diameter and circumference.
- 🔎 The independent variable (like diameter) is plotted on the x-axis, while the dependent variable (like circumference) is plotted on the y-axis.
- 📈 Proper scaling of axes is essential for accurate graph representation, ensuring data points use at least 75% of the graph space.
- ✨ A scatter plot shows data points but doesn't require connecting dots—it's the trend we're interested in.
- 📏 The line of best fit helps visualize a potential linear relationship in the data, modeled by y = mx + b.
- 🧑🏫 The slope (m) is the change in the dependent variable over the independent variable, here calculated as 3.
- 🟢 In this example, the relationship between circumference and diameter follows the equation C = 3D, which aligns with the formula C = πD.
- 💡 This graphing exercise illustrates basic principles of data visualization, important for physics studies throughout the year.
Q & A
Why do we graph data in physics?
-In physics, we graph data to find the relationship between two quantities. It helps us understand how changes in one variable affect another.
What is the independent variable in the given lab example?
-The independent variable in the example is the diameter of the round objects, measured in centimeters. It is placed on the x-axis of the graph.
What is the dependent variable in the given lab example?
-The dependent variable is the circumference of the round objects, measured in centimeters, and it is placed on the y-axis of the graph.
What should the title of the graph in physics look like?
-The title of the graph in physics should follow the format 'y versus x,' indicating the relationship between the two variables. In this case, the title is 'Circumference versus Diameter.'
How should the axes be labeled on a graph?
-The axes should be labeled with the physical quantity (like diameter or circumference) followed by the unit of measurement in parentheses, such as 'Diameter (cm)' and 'Circumference (cm)'.
What is the general rule for scaling a graph?
-A good rule of thumb is that the graph should take up at least 75% of the available space. If it doesn’t, you should reconsider the scaling.
Why should you not connect the dots in a scatter plot?
-In a scatter plot, you should not connect the dots because you are looking for a trend rather than exact connections between the data points. There may be experimental uncertainty, so a line of best fit is preferred.
What is the significance of a linear relationship in physics?
-A linear relationship in physics indicates that the data can be modeled using the equation y = mx + b, making the relationship between the variables straightforward and easy to analyze.
How do you calculate the slope of a graph?
-To calculate the slope, take two points from the line of best fit and use the formula: slope = (change in y) / (change in x). In this case, it’s the change in circumference divided by the change in diameter.
What was the slope found in this example, and why is it significant?
-The slope calculated was approximately 3, which is significant because it is close to the value of pi (3.14). This aligns with the known mathematical relationship between circumference and diameter, where C = πD.
Outlines
📊 Introduction to Graphing Data
The speaker introduces the topic of graphing data, emphasizing the purpose of graphing—discovering relationships between two quantities. The speaker shares a personal anecdote about their high school experience, explaining that while they previously thought graphs were just for getting good grades, the real reason is to visualize relationships. The example used is a data chart showing the relationship between the diameter and circumference of round objects. The speaker introduces key graphing concepts such as independent and dependent variables, and emphasizes the importance of labeling axes properly with units.
📐 Setting Up a Graph for Data
The speaker discusses how to set up a graph, focusing on the importance of scaling axes and titling graphs correctly. They explain the process of determining the appropriate scale for both the x-axis (diameter) and y-axis (circumference), ensuring the graph uses at least 75% of the space for better visualization. The speaker introduces the concept of a scatter plot and stresses not to connect the dots due to experimental uncertainty but to instead use a line of best fit. This line suggests a linear relationship, modeled by the equation y = mx + b.
🧮 Calculating Slope and Intercepts
The speaker delves into the details of defining and calculating slope, using circumference (C) as the y-variable and diameter (D) as the x-variable. They walk through the steps of calculating the slope (m) using two convenient points from the line of best fit, explaining the calculation in terms of change in circumference over change in diameter. They find the slope to be approximately 3, a unitless value. The y-intercept is identified as 0, meaning when the diameter is zero, so is the circumference.
📏 Deriving the Equation from the Graph
The final part of the explanation focuses on writing the equation from the graph, using the derived slope and intercept. The equation is simplified to C = 3D, and the speaker connects this back to the known formula for circumference (C = πD), pointing out that the slope of 3 is close to the value of π. This reinforces the relationship between circumference and diameter. The speaker concludes by summarizing the fundamental principles of graphing data and the significance of the results.
Mindmap
Keywords
💡Graphing data
💡Independent variable
💡Dependent variable
💡Scatter plot
💡Line of best fit
💡Slope
💡Y-intercept
💡Linear relationship
💡Math model
💡Experimental uncertainty
Highlights
Introduction to the concept of graphing data and its importance in finding relationships between two quantities.
Highlighting the reason why students graph data in AP Physics: to understand the relationship between variables.
Explanation of independent and dependent variables, with an example using diameter and circumference of round objects.
Introduction of the scatter plot and the importance of not connecting the dots due to experimental uncertainties.
Discussion on scaling the axes and tips for ensuring the graph occupies at least 75% of the available grid.
The identification of a linear relationship between diameter and circumference in the data set, making the analysis easier.
Introduction to the linear equation y = mx + b, and how it applies to the graph, where 'm' represents the slope and 'b' the y-intercept.
Assigning variables: Capital C for circumference and lowercase d for diameter, simplifying the mathematical model.
Step-by-step calculation of the slope using the formula ΔC/ΔD, explaining the choice of two points on the line of best fit.
The slope calculation resulting in a unitless value of 3, with the observation that the slope is approximately equal to π.
Defining the y-intercept as zero, as circumference equals zero when the diameter is zero.
Writing the final equation C = 3d, reflecting the relationship between circumference and diameter.
Recognizing that the slope value aligns with the mathematical relationship C = πd, linking the experimental result to theoretical geometry.
General tips on graphing, emphasizing how linear relationships simplify data analysis in physics.
Closing thoughts encouraging students to apply these graphing principles throughout their physics course.
Transcripts
hey everyone today we're going to talk
about graphing data so if we get started
here you know I want to think about a
question why do we graph data uh why
would we do this anyway when I was in
high school to be honest with you I
don't really think I knew this answer I
knew that any time I did a lab report
like I had to do graphs but I pretty
much just knew that sides teachers you
know liked graphs and so if you wanted a
good grade that's what you needed to do
but that's really not why we graph at
all so thing for a minute I mean why do
we graph data we graph to find a
relationship between two quantities and
in AP Physics the only reason why you
should want to graph data is if you
really want to see what the relationship
is between two quantities so as an
example to work with today I have a data
chart from a lab that could have been
done in class here and you can see the
first column here is diameter in
centimeters so we had students measure
the diameter of certain round objects in
the classroom and then measure the
circumference of those same round
objects and we have put it here in this
data chart now a couple of ideas about
data charts that are important this
first column here is the independent
variable so in this case we have
diameter and this is the quantity that
ends up on the x axis of your graph this
is the quantity that you determine the
second column of your data chart is
reserved for the dependent variable and
this is what we'll end up on our x-axis
so now if you take a look at the graph
to your right here we were given a grid
and I've set up the title when we think
about titles we want the title of your
graph and physics to always be y versus
X
so in this case it'll be circumference
versus diameter and you say I've just
used the symbols here one of the things
that takes me the longest amount of time
is scaling the axes so when I looked at
the x-axis I knew that this is where
diameter when and I have an axes label
here that says diameter and right next
to it in parentheses is the units with
which that physical quantity which was
measured so in this case centimeters
same thing on our y-axis here
circumference is the physical quantity
and then in parentheses you have
centimeters as the unit of measure when
i take a look at diameter the smallest
value is zero and the largest is 12 when
I counted the boxes here on the x-axis I
noticed that there were 12 boxes
relative to where the zero point was so
I just decided to have each of these
boxes count for one centimeter to do the
rescale for circumference was a little
more challenging the lowest value is 0
the highest is close to 38 when you look
at the graph here you don't have 40
squares or thirty eight squares so I
needed to think about how I could scale
the data and a good rule of thumb when
you're graphing is make sure that your
graph when you think about all the data
points here takes up at least
seventy-five percent of that graph if it
doesn't you can probably come up with a
method to scale it better so I have all
of the data points here and this is
what's called a scatter plot now I don't
want to connect the dots here because
I'm really just looking for a trend in
the data we talked about how this
information was collected and so there's
some experimental uncertainty when it
comes to real measurements here so I
definitely don't want to connect the
dots but i do want to line of best fit
between these points that you see here
and when i look at these points I get
the feeling that there this is probably
a linear relationship that we have
here and that's pretty exciting in
physics if you have a linear
relationship things are pretty easy when
we have a linear relationship we know
that we would be able to model that
graph in the form of y equals MX plus B
this is something you're probably very
familiar with because you probably dealt
with these straight lines over the years
the Y is going to be our Y quantity M is
the slope of the graph and notice that
we have a straight line anytime you have
a straight line your slope is constant
this particular straight line looks that
it looks like it has a positive slope X
will be our X quantity so in this case
diameter and finally be here is our y
intercept so now that we know that we
have a simple linear equation we can use
the graph here to write a specific
equation for this data but there's a few
steps that we need to follow so the
first thing that we want to do is define
the X&Y quantities so when I think about
this graph I have a why quantity all
right I have a wide quantity then in
this case is equal to circumference and
I don't really feel like writing
circumference down all of the time
especially if my main goal is to write a
math model for this particular
relationship so I'm would like to assign
a variable to circumference so I'm going
to call that capital C my x value
happens to be diameter and I know in
geometry mostly for diameter we use a
lowercase D so I'd like to assign a
lowercase D to that quantity here the
next step is to define and calculate the
slope I'm going to use the symbol M for
slope but just be aware that k can be
used for slope as well slope is always
change in the Y divided by change in the
X so in this case
change in the Y value means change in
the circumference and change in the x
value means change in the diameter when
I look at this definition of slope Delta
C over Delta D doesn't remind me of
another quantity so I'm just going to
leave that as Delta C over Delta D for
now but now let's calculate the slope
when you calculate the slope it's
important that when you're doing so
you're finding two points on the line of
best fit and not just grabbing two data
points so what I'm going to do is I'm
going to look for points on the graph
that are pretty convenient for me to
read so as an example here I might take
something like this right here I know
that this point is going to be eight
centimeters and the y value looked to be
right under the 25 so this is going to
be lets say about eight centimeters 24
centimeters for the XY coordinates now I
need any other point on this line and I
can choose any point on the line because
the slope of this line is constant ok so
what if I chose this point here is to be
right about halfway and so now I have
four centimeters and for the why I would
call that maybe 12 centimeters so let's
calculate this slope them on the top
here we have the difference in the Y
values so it'll be 24 centimeters minus
12 centimeters all divided by 8
centimeters / 4 centimeters this means i
end up with 12 centimeters on the top
and 4 centimeters on the bottom this is
interesting because a centimeter divided
by a centimeter is just one so I know
that whatever i do get for the slope
some unit list value so I know that the
slope is about three with no units ok so
now I've calculated my M so what's the
next step the next step will be to
define the y-intercepts and this is
pretty easy because all we need to do is
look at the y axis and find out what the
value is when x is equal to zero so when
x is equal to zero here circumference is
equal to 0 centimeters so I'm going to
say that be V is the symbol for the
y-intercept is equal to 0 centimeters
the last part of this is to finally
write the equation so I'm going to go
right over here to where it says y
equals MX plus B and now what I'd like
to do is use all of this information
we've collected to write a specific
equation for this relationship instead
of why i can substitute in circumference
c m the slope i've calculated as a
constant three or about three the x
value is diameter and be here is just
plus zero centimeters so if i want to
simplify that a bit i could definitely
do that i can simplify this to just
equals C equals three day because the
plus zero centimeters isn't important
and what's interesting about this three
here is that it is about PI right pi is
about 3.14 so you do get a slope that's
pretty much equal to pi and that does
make sense because before we even got
this started you probably knew that c is
equal to pi times D so does it make
sense that the quantity that we find the
relationship that we find is that
circumference is equal to about three
times the diameter of all the objects we
found right it definitely makes sense so
this is just an a small example of some
basic graphing principles that we'd like
to follow throughout the year I hope
that you found this helpful and I hope
that you have
a great day
Voir Plus de Vidéos Connexes
5.0 / 5 (0 votes)