Graphing
Physics concerns itself with finding functional relationships between physical variables. In class, you
will study some of the more important relationships that have been found to exist. In the lab, you
will perform experiments in order to confirm these relationships, and graphing is the tool that you will
use.
The first relationship under study here is that between circumference C and diameter d of a circle.
C = πd
(1)
You will use Equation 1 as written; it is easy to see here that the ratio
C
=π
d
is constant (the dimensionless constant π). Thus
C∝d
is the relationship. Theoretically, a plot of C vs. d should produce a straight line through the origin
with a slope of π. Experimentally, you will measure C and d for some circular objects and plot a graph
of C vs. d with the data. If this data produces a straight line through the origin then you will have
confirmed the relationship.
The second relationship is that between distance x and time t for an object moving with constant
acceleration a.
x=
1 2
at + vo t + xo
2
(2)
First, a simplification. If the object has no initial position or velocity this reduces to
x=
1 2
at
2
And now a complication. In the procedure you will be timing how long it takes the object to move a
set of distances that you choose. Therefore distance is the independent variable and we need to rewrite
the equation to reflect this.
2
t =
The ratio that is constant here is
1
2x
a
t2
2
=
x
a
and so
t2 ∝ x
is the relationship. Theoretically, a plot of t2 vs. x should produce a straight line through the origin
with a slope of 2/a. Experimentally, you will measure t for your chosen x and plot a graph of t2 vs.
x with the data. If this data produces a straight line through the origin then you will have confirmed
the relationship.
You will then calculate the rate of acceleration a. Since
experimental slope = slope of the line on your graph = theoretical slope =
2
a
then
a=
2
slope of the line on your graph
Apparatus
Disks, Rule, Measuring tape, Dynamics track and support, End stop, Meter stick, Masking tape,
Low-friction cart, Stopwatch.
Procedure
1. Using the rule and measuring tape, measure the diameter and circumference of each of the disks
provided. You may make each measurement once only but someone else should verify each
measurement to ensure good data. Record these values on the Data Sheet.
2. Plot a graph of Circumference vs. Diameter.
3. Do the remaining calculations.
4. The apparatus used to investigate the rerlationship between distance and time is shown in the
figure below. It consists of a long dynamics track elevated at one end; at the bottom of the track
is an end stop and no-bounce pad to gently stop the cart after accelerating down the track.
5. Adjust the height of the track such that the upper end is about 15cm above the table.
6. Choose four distances over which you will time the cart; go no smaller than about 25.0cm nor
larger than 175.0cm and make them roughly equally spaced between your extremes. Use the
masking tape to mark each on the side of the track if you want a visual reference (do not mark
on the track with a pen or pencil).
7. Release the cart from rest three times at each of your points and time how long it takes for the
cart to reach the bottom of the track.
8. Calculate the average time at each distance and the square of this average time.
9. Plot a graph of the Square of the Time vs. Distance Traveled.
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10. Do the remaining calculations.
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Data Sheet
d (cm)
C (cm)
Theoretical slope of line
Experimental slope of line
% Error
x (cm)
t (s)
t̄ (s)
t̄
2
(s2 )
Experimental slope of line (s2 /cm)
Experimental acceleration (cm/s2 )
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Analysis
1. What relationship did you test between circumference and diameter? Does your graph confirm
this relationship? If yes, then how; if not, then why not?
2. In the first procedure, why did you measure the circumferences; i.e., why can you not just use
Equation 1 to calculate the circumferences after measuring the diameters?
3. What relationship did you test between distance and time? Does your graph confirm this relationship? If yes, then how; if not, then why not?
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Pre-Lab: Graphing
Name
Section
Answer the questions at the bottom of this sheet, below the line (only) - continue on the back if you
need more room. Any calculations should be shown in full.
1. Read the lab thoroughly; check the lab manual for any additional information.
2. In this experiment, what is the relationship you are testing between circumference and diameter?
3. In this experiment, what is the relationship you are testing between distance and time?
4. What is the theoretical slope of the line on a graph plotted to confirm the relationship in Question
3?
5. How are graphs plotted in the physics lab?
6. How do you know whether or not your data fits a straight line through the origin?
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