In your lab notebook, start on a fresh page and write your name and your partners’ names
(first AND last) on the top right corner of the page. Write the title of the lab clearly, then
below it, hand-write the objectives section. As always, skip a line between your written
lines. One new section this week is “Relevant equations”; please follow the instructions in
that section as to what to include in your notebook. The other new section is “Prelab
questions” and I will grade them as I assess your notebook during class, so write the
answers (and include the questions) in your notebook.
You may cut and paste (typically tape works better than glue to hold papers in the notebook)
the equipment (including the setup photo) and procedure sections. Set up the data tables
for the data section. Complete all this prior to the start of lab on Thursday; I will check your
lab notebook at that time.
The writeup to this lab is due Friday, January 22, though you may submit it early. The
writeup consists of a photocopy of your lab notebook pages for Lab 2, from the title page to
the conclusion section.
Physics 221
Lab 2 – The measurement of “g”
Objectives:
• Measurement of the speed of a standard “frictionless” cart on a ramp using three different
timing devices
• Performing data analysis on these measurements, including the determination of
uncertainties
• Propagating uncertainty in a simple ratio
Introduction and relevant equations used:
The term “g” appears in many kinetics equations, especially those to do with ballistic
trajectories. “g” is simply represents the acceleration of an object due to the Earth’s gravity
and has a value of approximately 9.8 m/s2. In this lab, you and your team are going to
measure the motion of a cart down a ramp, varying the angle of the ramp, and then calculate
different values for “g” to a high degree of precision.
ax
ay

acar
t
Galileo Galilei tried to measure g, according to legend, by
dropping cannon balls off of the Tower of Pisa. But g is a
relatively large number and dropped objects speed up
quickly. Thus, we’re going to employ an indirect way of
calculating g by measuring the acceleration of a cart down
a ramp due to gravity, which will be a smaller (and more
measureable) value than a free-fall drop, then correct for
the ramp angle.
In the diagram above , you can calculate acart with a measurement method from Lab 1 and a
ramp set up with a cart. You can measure the angle of the ramp. Using a trig formula, you
should be able to calculate the components of the acceleration, that is, ax and ay from these
two pieces of data. Furthermore, g will be one of these two components.
In your lab notebook, as preparation, draw the above vector triangle (with the sides and
angle labeled). Label the component that is equal to g. Finally, write three formulas:
• the equation that connects the angle of the ramp and the acceleration of the cart to g.
• the equation that connects the displacements measured to instantaneous velocity.
• the equation that connects the instantaneous velocities to the acceleration of the cart.
Potential equipment:
(strike out those pieces of equipment that you did not use and add any pieces of equipment
that you used that aren’t listed below):
2.3-meter metal ramp
Collision cart that fits on the ramp
Popsicle stick “targets”
Go!Motion sonic rangefinder
Stopwatch
Two photogates
LabQuest 2 datalogger
Laptop with Logger Pro software
Take a photo of your final setup with all measuring devices in place to include in your lab
notebook here.
Procedure (work in teams of four):
1. Decide on the angle measurement app or device you are going to use; make sure it gives
reasonably accurate angles by comparing it against the levels available in class.
2. Set the ramp such that it is at an angle (not too steep; should be less than 30° initially) and
the numbers get bigger downhill on the yellow rule.
3. Use tape to mark the “start” line and the “finish” line; avoid blocking the cart’s wheels or
path with tape. In the data section, record the length of the distance between the tape lines, if
this measurement will not be made any other way.
4. Set up the measurement method you believe will give you the needed data to calculate g.
5. Set up the cart on the ramp; it should run smoothly and easily (no bumps or sudden
slowdowns), and trigger the measurement method as needed. Remember that the cart should
be held in place until it is let go and the measurements taken; do not give the cart a “push”
down the ramp. Adjust the sampling time to give you at least ten useful data points (see
question 1).
6. Run the experiment at least five times, each run at a different angle, each time recording
the time and displacement measurements. This should end up being a two-column (or tworow) table with many entries.
7. Once you have obtained one set of time/displacement data, calculate the instantaneous
velocities for that run. Then, using those instantaneous velocities, calculate the
instantaneous accelerations of the cart as it went down the track.
8. Assess whether that run yielded good data (see question 4); if it did, then move on to the
next run and repeat, until you have five “good” runs.
9. Make sure all equipment is returned to their appropriate places, and that all members of
your team have all the data files/photographs they need to complete the lab write-up.
Prelab questions
1. From last week, what’s the smallest time slice (sampling time) possible on the
photogates? On the sonic rangefinder?
2. What method of measurement (photogates or sonic rangefinder) are you going to use?
What are the advantages of this piece of equipment over the other? What are the
disadvantages, if any?
3. Why is it better to have many time and displacement measurements, instead of just an
initial and final set of measurements? What type of acceleration do you want to calculate
from the measurements, ultimately?
Data section
Start with the calibration of your angle-measuring application or device. Include the
comparison the levels in the classroom. State whether you believe your app/device to be
accurate and include an estimate of uncertainty (in other words, what precision do you earn?).
USE TABLE FORMAT for repeated trial experiments; always include units and other
identifying information.
Run 1: Angle of ramp off of horizontal: ______°
time (s)
uncertainty
in time (s)
displacement
(m)
uncertainty
in disp. (s)
Note: this may be combined with the analysis table below for each run; please label things
clearly. It is also okay to paste in Excel or other spreadsheet tables. Further note: in my
format, the time between two adjacent columns will be called an interval.
Analysis section
Using the equations from the revelant equations section, fill in the velocity and acceleration
for each interval. If this calculation is easier to perform on a spreadsheet program, feel free
to use it.
Instantaneous
velocity
(units?)
uncertainty in
velocity
(units?)
Instantaneous
acceleration
(units?)
uncertainty in
acceleration
(units?)
Propagate the uncertainty in time and displacement as you did last week to get an
uncertainty in velocity for each interval. Perform a similar calculation to derive an
uncertainty in acceleration.
Illustrate these tables by plotting three graphs: displacement vs. time, velocity vs. time,
and acceleration vs. time. Make sure to follow all the rules of good graph-making given out
in class (stuff like label axes, tick mark consistency, etc.). For the uncertainties, use
uncertainty ranges (bars).
Using your criterion for discounting measured data in question 4, throw out those data
points, then calculate a mean acceleration for each run. Using the formula for standard
deviation from last week, calculate the standard deviation in the acceleration for a run.
Write the result of each run as mean acceleration ± standard deviation (match precision),
followed by units.
Using a formula from relevant equations, calculate g for each run.
Decide whether to use the standard deviation or the uncertainty in acceleration for the next
calculation (hint: pick the larger one).
Using the formula for calculating uncertainties from last week (given below), determine the
uncertainty in g for each run. Write the formula explicitly evaluating the partial derivatives
so that there are only variables for acceleration and angle.
2
𝜕𝐶
𝜕𝐶
𝑢(𝐶) = √(𝑢(𝐴) ( )) + (𝑢(𝐵) ( ))
𝜕𝐴
𝜕𝐵
2
Discussion and results section
Typically, this is where the results and data analysis are explained, especially sources and
magnitudes of error; however, for this lab, I am going to guide you by having you answer
the following questions (number the answers, please).
Question 4: How can you tell if a particular run is “good”, based on the instantaneous
accelerations you calculated? Can you discount some of the data that gave inconsistent
accelerations? If so, what is the valid reason for doing this?
Question 5: Why aren’t the uncertainties in velocity necessarily the same for each interval?
Question 6: Report a mean g value and its associated standard deviation. Don’t forget the
units, and do match precision. Make it a sentence by starting: “Our value of the acceleration
due to gravity, g, is…”
Question 7: Report a percent error of your mean g value compared to the accepted value of
g found in the inside cover of your book. Percent error is always calculated:
𝑝𝑒𝑟𝑐𝑒𝑛𝑡 𝑒𝑟𝑟𝑜𝑟 =
|𝑦𝑜𝑢𝑟 𝑣𝑎𝑙𝑢𝑒 – 𝑎𝑐𝑐𝑒𝑝𝑡𝑒𝑑 𝑣𝑎𝑙𝑢𝑒|
× 100%
𝑎𝑐𝑐𝑒𝑝𝑡𝑒𝑑 𝑣𝑎𝑙𝑢𝑒
Conclusion section
Again, typically, this is where you summarize the main findings of the experiment, state
your assumptions which limit the results, and assess sources of error in the experiment.
Here, too, I will guide you by asking questions:
Question 8: Were the instantaneous velocities and accelerations actually instantaneous?
What hindered you from calculating truly instantaneous velocities and accelerations?
Question 9: Typically, the mean g calculated in this fashion is lower than the accepted
value. Even if this is not true for your measurements, explain a simple reason why this
might be so. (Incidentally, this is definitely a source of systematic error).
Question 10: Did the uncertainty in g increase or decrease as the angle of the ramp
increased? Even if it did not for you, give a reason why this might be a typical result.