AP Physics 1: Introduction 7: Basic Graphing

Summary

This 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

00:00

📊 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.

05:02

📐 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.

10:05

🧮 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.

Keywords

💡Graphing data

Graphing data refers to the visual representation of data in the form of a graph, such as a scatter plot or line graph. In the video, graphing is explained as a method to explore relationships between two quantities. The speaker emphasizes that graphing is essential in physics to understand how one variable changes in relation to another, like the relationship between diameter and circumference.

💡Independent variable

The independent variable is the quantity that is controlled or determined by the experimenter. It is plotted on the x-axis of a graph. In the video, the diameter of objects is used as the independent variable, as it was measured and controlled in the lab experiment.

💡Dependent variable

The dependent variable is the quantity that changes in response to the independent variable and is plotted on the y-axis. In the video, circumference is the dependent variable, as it depends on the diameter of the objects being measured. The purpose of graphing is to observe how changes in the independent variable affect the dependent variable.

💡Scatter plot

A scatter plot is a type of graph that displays data points on a grid, showing the relationship between two variables without connecting the points. In the video, the speaker creates a scatter plot with the diameter on the x-axis and circumference on the y-axis, which helps to visualize the linear relationship between these two variables.

💡Line of best fit

The line of best fit is a straight line drawn through the data points on a scatter plot that best represents the trend in the data. The speaker explains that it’s not necessary to connect individual points in a scatter plot but instead to use a line of best fit to identify the overall trend, which in this case is linear.

💡Slope

The slope of a line on a graph represents the rate of change between the dependent and independent variables. It is calculated as the change in the y-value divided by the change in the x-value. In the video, the slope of the line of best fit is calculated as approximately 3, indicating that for every unit increase in diameter, the circumference increases by about 3 units. The slope also hints at the mathematical relationship between diameter and circumference (C = πD).

💡Y-intercept

The y-intercept is the point where the line crosses the y-axis, representing the value of the dependent variable when the independent variable is zero. In the video, the speaker identifies that the y-intercept is 0, which makes sense because if the diameter of an object is 0, its circumference would also be 0.

💡Linear relationship

A linear relationship is a direct, proportional relationship between two variables, meaning the change in one variable causes a consistent change in the other. In the video, the relationship between diameter and circumference is shown to be linear, following the equation C = 3D, which is a simplified form of the well-known formula C = πD.

💡Math model

A math model is an equation that represents the relationship between variables. In the video, the speaker explains how to derive a mathematical model from the graph by identifying the slope and y-intercept, resulting in the equation C = 3D. This equation models the relationship between circumference and diameter based on the data collected.

💡Experimental uncertainty

Experimental uncertainty refers to the inherent inaccuracy or variability in measurements due to limitations in the experiment. In the video, the speaker mentions that real measurements have experimental uncertainty, which is why the data points do not form a perfect line and why the line of best fit is used instead of connecting the individual points.

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