Physics I Laboratory Online
Laboratory Methods
Name
Date
1. Objective
? Practice graphing data and interpreting the results,
? Perform averages and standard deviations of measures.
2. Apparatus
You will need math software (such as Excel, Matlab, Mathematica, GNU Octave), a length
measuring device (ruler, measuring tape, or meter stick; metric is preferred but not required),
stopwatch (smart phone would work well), pendulum from the first experiment, and graphing
paper. If you do not have graphing paper, you can easily find and print it from online.
3. Procedures
I. Graphing and Interpreting Linear Data
A. Make a plot of the following data on a piece of graph paper, plotting position (x) vs
time (t). IF you do not have graphing paper you can download some samples. You
should label the axes with proper units. Draw a best fit line through the data by
hand. The line should be as close to as many data points as possible. Determine
the slope (m) and y-intercept (b) of the line. Write the equation of the line using
slope intercept form, x(t) = mt + b.
t(s) x(m)
0 10
1 21
2 29
3 41
4 49
5 60
m = ____________________, b = ______________________
Write an expression for x(t) in slope-intercept form, x(t) = mt+b
x(t) = __________________________________
2
B. Repeat the graphing exercise, but this time use your favorite graphing tool, Matlab,
Excel, Mathematica, graphing calculator, etc. Enter the data into lists, plot one list
vs the other list, then have the program calculate the best linear fit to the data. Also
include the Root Mean Square Error or some other measure of how good the fit is.
Include a printout (or image) of your work with this report.
Math Tool Used _____________________
Results of the linear regression:
Slope, m = _______________________________
Intercept, b = _______________________________
RMSE = __________________________________
II. Graphing and Interpreting Data
A. Pendulum Period vs Length
You will measure the period of pendulums of various lengths, analyze the results and determine
the dependence Period has on the Length of the pendulum.
1. For the length and mass used in your previous lab (~1 m in length) you will use the average
Period calculated from that data. If you do not have the pendulum from your last lab, make
a new one and determine the period as described below.
2. Next you will construct a second pendulum with a length of approximately 80 cm (31.5 in)
using the same mass above. You may tie the 1 m pendulum a bit shorter to make this
pendulum. Again it does not need to be exact, just close to 80 cm.
3. You will measure the length of the pendulum from where the pendulum is supported (pivot)
to the center of the bob. Measure your pendulum and record its length in Table 1.
4. Pull the bob ~10? from the equilibrium position
5. Let the pendulum swing several times until it swings steadily. Get the stop watch ready and
when the bob swings to one side and comes to a stop at the top of the swing start the stop
watch.
6. You will measure the time for 10 cycles. One cycle is the time it takes to go from where you
released the bob to the opposite side of the swing and then comes back to where it started.
Remember you do not count one until the bob comes back to you for the first time.
3
7. At the end of the tenth cycle stop the stop watch.
8. Record the total time t for ten cycles in seconds, to the nearest 100th of a second. Record
these values in the proper columns in your data table. You will collect only one set of time
data here.
9. Construct other pendulums with lengths of approximately 60 cm (23.6 in), 40 cm (15.7 in) and
20 cm (7.9 in) using the same mass and repeat steps 3 8 for each length.
10. Determine the period of oscillation, T, for each Length and place the result in your data table.
11. Using your favorite graphing tool (such as Excel), plot the periods versus the length. (include
the graph when you turn in your report)
12. Fit a power law to your data (of the form)
y = aLb
(1)
where a is the power law coefficient, L is the length of the pendulum, and b is the power.
Based on your fit, what are a and b?
a = _________________ b =__________________
13. Comparing equation 1 above to ? =
2?
??
?
1/2
(equation for the period of a pendulum), how
does your value of a compare to 2?
??
? Using your value of a from the power law fit, what
value of g, the acceleration due to gravity, can you infer? Show your work below or include
it with your report.
14. Calculate the % error of this experimental value of g with the accepted value of 9.81 m/s2
.
% ????? =
?????9.81
9.81
*100
15. Is b what you expected? Compare to ½.
4
Table 1. Pendulum Period versus Length
Approximate lengths Actual Length
(m)
t (10 cycles) T (Period)
~1 m
~0.8 m
~0.6 m
~0.4 m
~0.2 m
III. Simple Statistics for the Physics Laboratory
A. Multiple Measures: Mean, Standard Deviation, and Mean Error.
If we take a number of measurements, n, of some physical quantity x then the best
experimental value we can obtain for x is the arithmetic mean (we use x with a bar over it to
indicate average value of x) of the set of measurements. The capitol sigma (?) is a shorthand
known as the summation symbolits meaning is illustrated in the expressions below. This
mean (also called the average) value is defined as
Thus, the mean is the sum of all the measures made divided by the number of measures made.
Another way of describing the data is by how much spread there is in our measures. This
quantity, which gives us a measure of the precision of our data, is the standard deviation, ?,
defined as
5
The standard deviation has the property that if we were to make one more measurement of
the quantity x, there would be a 68% probability that the measurement would be within one ?
of the mean value, i.e. it would fall between ? ? and ? + ?. In the figure below, two sets of
data are presented, one with a small deviation and one with a large deviation.
Figure 1. Two sets of data are presented with each color representing +/-1? (red),
+/- 2? (green), and +/- 3? (cyan) deviation from the mean.
So we can determine the mean of our measures and get a sense of how spread our measures
are, but how well do we know the mean. The error in our mean (called the standard error) is
given by ? ? =
?
??
. Thus, the more times you measure a quantity, the more accurate the mean
of those samples.
You will make 10 good measures of your walking stride length (length of one step) measured
from the tip of the toe of one foot to the tip of the toe of the other foot (or other identifiable,
repeatable location). You will need to develop a method of marking and measuring your
strides. Make sure you are measuring your walking stride, not just a step. Describe your
method below. A puddle of water to paint your steps, printer paper laid out to leave
footprints on, or walk on a sandy/dirty surface are a few possibilities to help make your
measurements. All 10 strides do not need to be made at once, just make sure you are walking
when they are made. Once you have collected 10 good measures of your stride, compute their
mean, standard deviation, and standard error.
Describe how you measured your stride here:
6
Stride Lengths:
___________________, __________________, __________________
___________________, __________________, ___________________
___________________, __________________, ___________________
___________________
In the space below, please show the work used to determine the numerical values for the
mean, standard deviation, and standard error. Do this by hand. Remember to include units.
In this space show how you determined the average for these 10 readings:
Average Stride Length= ____________________________________________
In this space show your work of how you determined the standard deviation for these 10
readings.
Standard Deviation in Mass Readings = _______________________________
In this space show how you determined the standard error for these 10 readings.
Standard Error in Mean of stride length = _____________________
7
Now use your favorite math tool to compute the average and standard deviation of the 10
values.
Math Tool Used__________________________
Results: Average = ________________________
Standard Deviation = _________________
Standard error of mean = _________________
