GRAPHING TECHNIQUES
(3 labs: labgraph1 , labgraph2, labgraph3 , labgraph4 )
(bouncing ball , density of water, linear density of wires, walking speed)
0) Read: I always give hints. Try first WITHOUT looking at them. Then use the hints to check your work.
Suggestions: copy and paste the exercises in a word document. Format them. Leave blanks for your answers.
you can type the answers if you like. If not, you can print the document and write the answers. I don't collect HW but I do call on students. We will go over the HW in class. You need to work you assignments thoroughly as our Next quiz = exercises from HW (without the hint) . We have a 20 minutes quiz every cycle.
If you really LOVE Physics, see me for extra assignments on line !
1) Read and complete: In Physics, the relationship between the experimental values is shown using a technique using graphing.
The independent variable is always plotted along the x axis of the graph, and the dependent value is plotted along the y axis of the graph.
Example: Say a car has an average speed of 40mph. Let's the time (in hr) be the independent variable and the distance (mi) covered
the depended variable. This means, the distance covered depends on the time.
Usually, the first thing to do is to build a (x,y) table. Fill the following table:
| x= time | y= distance |
| 1 | |
| 2 | |
| 3 |
This is called a linear relationship. WE say y is proportional to x.
It is easy to plot the points on a graph. Since the relationship is linear, we trace a straight line through the points:
Plot the points (x,y). !! Give a title to the graph, label the axes,
think about the right scale. !!
You should use your own graph paper, in that case,trace the first quadrant as distance and time are positive.
GRAPH PAPER
The slope tells you about the steepness of the line.
slope = rise/run = change in y / change in x. In that case the slope = _______ mi/hr
The slope is a rate. It tells you by how much the speed increases every second.
The equation of the line is y = ___ x.
watch this movie in class. Choose the segment called: graphing relationships (10:59)
http://player.discoveryeducation.com/index.cfm?guidAssetId=BECB9850-FB39-4BCD-BBD0-16413F95F261
(ask me for login and password).
The movie tells about how graph can be used to derive relationships between
2 variables and the meaning of a slope.
2) REad and fill. In the previous case, the line go through (0,0) and the equation of the line is in the form y = m x where m is the slope. (In Math you call that the slope-intercept form).
It can be that the straight line does not go through (0,0)
Imagine you take Summer classes in an university. The registration fee is $500 (only paid once, independent of the number of credits you take) and then you pay $100 per credit. If you register and take no credit, you still pay $500.
If you take 1 credit, you pay _____ + ______ = ________
Instead, if you take 2 credits, you pay _______ + 2 (_______) = _____________
and so forth.
The independent variable x is the __________________ and the dependent variable y is ________________----
Fill the table :
| x = number of credits | y = tuition |
| 0 | 500 |
| -------------- | --------- |
| ------------- | -------- |
| -------------------- | --------- |
You see that the equation of the line that gives the relationship between the tuition (y) and the number of credits (x) is:
y = ____ x + _____. This is again a linear relationship.
Plot the points (x,y). !! Give a title to the graph, label the axes, think about the right scale. !!
(hint: along the y-axis 1 unit = 500 , along the x-axis 1 unit = 1 )
Trace a line through the points. The y-intercept is the intersection between the line and the y-axis.
GRAPH PAPER
The equation of the line is y = ______________. The slope is _________
(rise/run = change in y / change in x)
The slope tells you by how much the price increases each time you add one credit.
The y-intercept is ________. The y-intercept is where the line cross the y-axis.
( That is : x = 0 , in that case you just register and take no credit)
The equation of the line is y = ____ x + _____. The form is called slope-intercept.
3) do the following lab: labgraph1
you need bouncing balls.
In Physics, even if you expect a linear relationship between 2 variables, the graph will not be a perfect
line if the data are collected during an experiment. Some inaccuracy occurs because of human errors.
First you do a scatter plot. (plot the points). If you see a linear trend, the relation is linear. Trace your best fit line. Here is a nice video that shows you the idea:
4) Students are given a toy car to study uniform motion.
During the experiment they record the distance d(m)
covered by the car as well as the time t(s). Their measures are recorded below:
| x= time (s) | y= distance (m) |
| 1.0 | 0.32 |
| 2.0 | 0.80 |
| 3.0 | 0.95 |
| 4.0 | 1.18 |
| 5 | 1.6 |
A) The students try to find out if there is a relationship between the distance and the time. the time is the ____________ variable (independent ? dependent ?)
the distance is the __________ variable.
B) To find out what kind of relationship exist between the variables, you need to make a graph
(so get a graph paper). NEW: you can use your TI for this one. See this MOVIE to find out how !!
Trace the x-y axis - Find the right scale
(example: 1 unit on x-axis = 0.1 and 1 unit on y axis = 1 )
plot the points. Don't connect the dot. Label the axis. Give a title to your graph
Is this a linear relationship (does it look like a line ? )
C) Find the best fit line. don't connect the dots. go through (0,0)
D) Find the slope (see the lab for instructions). slope = _____ m/s
E) Can you find the relationship between y and x ? y = _____ x
since y is the distance and x the time, you can write d = _____ t.
F) the unit of the slope is m/s. The slope represent therefore the __________ of the car. (m/s or mph is the unit of _____)
The slope means that the car will cover ________ meters in one second.
5) lab fitting a line to a set (labgraph2) of data to make prediction
6) During an experiment, a student measured the mass of 10 cm3 of alcohol.
The student then measured the mass of 20cm3 of alcohol. In this way, the
data were collected: (if you know how to use the TI, use it, draw what you get on a separate paper)
| x= volume (cm3) | y = mass (g) |
| 0 | 0 |
| 10 | 8 |
| 20 | 16 |
| 30 | 24 |
| 40 | 32 |
| 50 | 40 |
(hint: 1 unit along x = 10 , 1 unit along y = 5 try)
B) trace the best fit line (going through the origin ). slope = ______g/cm3
The slope represents the ________ of alcohol. (mass per unit volume is the _______)
It means that 1 unit volume of 1 cm3 has a mass of ________
It means: each time you increase the volume by 1 cm3,
you increase the mass by __________.
C) Find the relationship between y (the mass) and x (the volume). y = ______ x.
D) CAn you use your equation to predict the mass of x= 35cm3 of alcohol ?
y = mass = __________. (hint: use the relation between y and x)
Learn how to find the best fit line and to use the TI to do the job for you:
labgraph3 -----
7) During a class demonstration, a physics instructor placed a 1kg mass on a horizontal table was nearly frictionless.
The instructor then applied various horizontal forces F to the mass and measured the acceleration a for each force applies.
The acceleration tells you by how much the speed increases every second.. That is how fast the mass get faster.
So the input x (or independent variable) is the force F (N) = x and the output
(or dependent variable ) is the acceleration a (m/s/s) = y. The results of the experiment are shown here:
| x=force (N) | y=acceleration (m/s/s) |
| 5 | 4.9 |
| 10 | 9.8 |
| 15 | 15 |
| 20 | 20.1 |
| 25 | 25 |
| 30 | 29.9 |
A) Graph acceleration vs force : plot the points. don't connect the dots. Don't forget to label the axis and to give a title to the graph. Find the best fit line.
(you can use the TI if you know how)
B) Find the slope of the line. slope = _____ m/s2 / N
extra credits: a Newton = N = kg . m/s2. Find now the new unit of the slope. (m/s2 / kg . m/s2 )
The slope represent the inverse of ______, the unit is ________ in the SI system.
8) A) An experiment is done by students. They measure the weight (y) of different masses (x). They report their measures in the following table:
| x= mass (kg) | y=weight (N) |
| 0 | 0 |
| 5 | 50N |
| 15 | 148N |
| 20 | 203N |
| 25 | 249N |
| 30 | 303 N |
Plot the points on a graph and trace the best fit line. don't forget to label the axes, give
a title ... (hint: 1 unit x = 5 and 1 unit y = 50 )
Estimate the slope of the line slope = _____N/kg
Is this number surprise you ?
On Earth, weight = mass x ________ ? (at sea level)
the equation of the line is therefore y = _______ x.
The slope represents the gravitational acceleration on EArth.
B) An astronaut conduct experiments on a distance planet to find the gravitational acceleration of the planet. His measurements are reported in the table:
| x= mass (kilograms) | y= weight (newton) |
| 0 | 0 |
| 15 | 106 |
| 20 | 141 |
| 25 | 179 |
| 30 | 216 |
| 35 | 249 |
Help the astronaut to find the acceleration of this new planet. Suppose the relationship between weight and mass is linear. y = m x. You are looking for m. (the slope of the line)
10) During a unit in electricity, students built a circuit including a generator (like a variable battery) and a resistor. Using the generator they generate currents of different magnitudes (independent variable x) and record the tension (difference of voltage) across the resistor (dependent variable). They recorded the data collected in the following table.
| y= Tension across the resistor (volts) | 0 | 1.2 | 2.5 | 4.8 | 10.2 | 14.8 | 20.5 | 27.0 |
| x= Current (amps) | 0 | .025 | .05 | .1 | .2 | .3 | .4 | .6 |
Find the relationship (linear) between the tension y and the the current x.
(you can use the TI). y = mx . you need to find the slope. if you use the TI you should get
b very small. b is actually 0. When x = 0, y = 0. (zero energy = zero current)
y = ____ x .
In Physics the symbol of a tension is U and the symbol for current is I.
The relationship is linear. U = R I. This is called Ohm's law/ R is the resistance of the circuit.
In your experiment R = ____ Ohms.
11) A number of different temperature scales have been devised, two popular choices being the Celsius (formly centigrade) and Fahrenheit scales. Historically, both scales where defined by assigning 2 temperature points on the scale arbitrary and then dividing the distance between them equally spaced intervals.
On a Fahrenheit scale, the water boils at 212F and freezes at 32F.
On a Celsius scale, the water boils at 100 C and freezes at 0C.
The relationship between Fahrenheit and Celsius is linear.
If you have Fahrenheit = y and Celsius = x then y = mx + b or F = mC + b/
A) Find the linear relationship between F and C using your TI or using algebra
hint: The line of F versus C contains the 2 points:
( x=0, y= 32 ) and (x = 100 , y =212) . you can either :
A) use the TI: Enter these 2 point in STAT EDIT then find the equation of the line. (follow the steps)
using STAT CALC .
B) use algebra. You can also plug (x1,y1) and (x2,y2) in y= mx + b to solve for m and b. (you have a simple system of 2 variables)
C) use geometry. or you can use the point-slope formula (y - y1) = m (x - x1). with (x1,y1) either one of the 2 points and m the slope , m= (y2-y1) / (x2 -x1) First find the slope then plugthe slope m and (x1,y1), any of the 2 points, into your formula.
you find F = ___ C + _____
Write the slope as a fraction = _______ (on TI use key MATH 1:)
This is the way the formula is written in most book. (to go from decimal to fraction using the TI: use the MATH key
and then FRAC)
B) An healthy person has an oral temperature of 98.6F . What would this reading on the Celsius scale?
hint: F = 98.6 solve for C using your equation found in A
C) A "time and temperature" sign on a bank indicate the outdoor temperature is - 20C. Find the corresponding temperature on the Fahrenheit scale.
D) A room temperature in Europe is 19C , How much is that in F ?
12) In Physics we use another temperature scale called the Kelvin temperature scale. Named after the Scottish physicist Lord Kelvin. A temperature of 300K is said "three hundred kelvins ". The kelvin is the SI base unit for temperature. The relationship between kelvins and degrees Celsius is K = C + 273.15
Experiments with gases have shown that there iss a lowest possible temperature. This temperature is defined to be the zero point on the Kelvin scale (=0 K ). This is referred as the absolute value. At that temperature, the atoms
(molecules) stop moving and electricity can circulate without any resistance.
On the Celsius scale, the absolute zero is ______ C
(hint use C= K + 273.15)
The ice point (water freezes, 0C) occurs at _________ K on the kelvin scale.
The surface temperature of Sun is 6000 K. How much is that is degrees Celsius ? ___________
