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 thisprior 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.
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 diagramabove , you can calculateacart 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 rampStopwatch
Collision cart that fits on the rampTwophotogates
Popsicle stick “targets”LabQuest 2 datalogger
Go!Motion sonic rangefinderLaptop 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 two-row) 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? Whattype 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, thencalculate 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.
Discussionand 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:
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.