Materials
4 each photo gates with interface cables1 each Vernier calipers2 each pendulum masses, wood and brass4 each universal mounts1 each large ring stand and basePC, SW750 interface1 each metric rulerfoam padding
Goals
1) To collect consistent velocity-time data for the “free fall” of an object (in this case, a wood cylinder) that may be greatly affected by air resistance.
2) To collect consistent velocity-time data for the “free fall” of an object (in this case, a brass cylinder) that will be much less affected by air resistance.
3) To calculate standard deviation and apply it appropriately; to plot “error bars” on graphs using the standard deviation as the uncertainty estimate.
4) To determine by graphical means whether the acceleration experienced by a freely falling object is constant and, if so, to calculate the value of the constant acceleration.
Introduction
When an object is dropped from rest, its speed increases as it falls. Since the speed changes with time, acceleration must be present. In the experiment today you will be characterizing the motion of a freely falling body in detail. The concepts of the mean (or average) value and the standard deviation (an estimate of the uncertainty of the mean value) are also used in the analysis. [See the Uncertainty/Graphical Analysis Supplement at the back of the lab manual for details and definitions – or, ask your lab instructor.]Timing the Falling Object
Four infrared photo gates, each above the other, are sufficient to characterize the motion of a falling object reasonably well. Refer to the Infrared Photogate section of the Computer Tools Supplement at the back of the lab manual for specific instructions on connecting photo gates to the computer. Use Channel 1 on the SW 750 for the upper photogate and successively Channels 2, 3, and 4 for the lower photogates. You need to set up the Data Studio software to display “Velocity in Any Gate” and corresponding “Time in any Gate” (preferably in a table) as the object passes through each of the photo gates. Turn off any other “Data” quantities that are defaulted by the software. Note: These data options only appear for whichever gate you attached to the interface second. (That’s because the computer only realizes that it will need to integrate data from multiple gates, at the time you attach the second one.)In order for the computer to generate velocity data, you will need to input the value of the falling cylinder’s diameter under “flag length” in the “constants” tab of the “Experiment Setup” window in Data Studio.There is a default value in the software so you will not get an error message to remind you about this important step. If you fail to enter the correct size, all of your velocity results will be incorrect. The computer will calculate the velocity by dividing this “flag length” by the amount of time the photogate’s beam was interrupted.
Flag Length with Vernier Caliper Accurate measurement of the cylinder’s diameter can be made using the Vernier caliper provided at your station. The Verier calipers have a second, sliding scale attached to the bottom ‘jaw’ of the caliper. Note: The zero “0” on the sliding scale lines up with the value of the measurement on the fixed scale, accurate down to the millimeter level. Measure from the “0” not from the edge of the jaw! Be sure to round this measurement down (never up) to the nearest millimeter (you’ll se why after the next paragraph.)The rest of the sliding scale may then be used to get a reading accurate to one tenth of a millimeter, as follows: Without letting the scale move, visually scan the sliding scale, looking for the hash mark on the sliding scale which most nearly lines up to the fixed scale hash mark above it. This value on the sliding scale is the number of tenths of a millimeter you should add to you initial result. For an illustration, see the figure below and/or visit the following url: http://www.upscale.utoronto.ca/PVB/Harrison/Vernier/Vernier.html
The Vernier caliper above reads “3.58mm” as each hash on the sliding scale is 0.02mm.
The Experiment
Part 1A wood cylinder (wooden pendulum mass, minus the hanging hook screw) is the object to be dropped through the photo gates. Take some care to align the photo gates such that the cylinder can fall without hitting any of them. The cylinder should be released from rest at a distance of 0.15-0.25 m above the first photo gate. Drop the cylinder so that the axis of the cylinder is horizontal and is nearly perpendicular to the light beam of the photo gates. Your lab instructor will demonstrate the technique for you. It is important that the cylinder fall without wobbling or tumbling significantly. Experiment to find a suitable technique. No technique is foolproof, so you will have to sort out the runs where nothing goes wrong. Seek out your lab instructor for suggestions if you are having trouble. Expect to drop it several times before getting a “clean” run without rotating significantly or hitting a photo gate.For the remainder of the experiment, keep the distance between the drop point and the top-most photo gate fixed. Failure to do so will negate the usefulness of your data, as multiple trials cannot be averaged together, unless they were performed under the same conditions.The position of the lower photo gates should be roughly equally spaced below the upper photogate with the lowest photo gate situated very near to the end of the mounting rod. To reduce errors get at least three consistent “clean” runs. Look for consistency in your data by comparing not only the average velocities through each gate but also the time intervals from the first gate to each of the other gates. If one drop of the cylinder is inconsistent with the others, it may very well be that it tumbled too much or hit a photo gate or even missed breaking one of the beams.Note: If the cylinder is rotated as it passes through a photo gate, then it will take longer to pass through the beam because it appears larger than just its diameter; the longer time interval is then interpreted by the computer as if the cylinder is moving slower than it really was. Appoint one member of your lab group as the “referee” to watch each drop carefully and to determine which runs should be counted. The person dropping the cylinder is not in a good position to determine whether the cylindrical axis of the cylinder is rotated from the horizontal as it falls. Unless an obvious problem occurs, such as the cylinder hitting a photogate or the like, you should record the data. Hopefully the “good” runs will give consistent velocities varying by only a few percent. If you print out a computer generated data table, make sure that with the help of the “referee” you cross out any bad runs and include short notations as to the problems with those runs. Calculate the mean (or average) and standard deviation of the speeds through each photo gate. Refer to the Uncertainty/Graphical Analysis Supplement at the end of the lab manual – or ask your lab instructor. The averages and standard deviations of the elapsed times from the first photo gate to each successive photo gate also need to be calculated. These standard deviations will be used to determine error estimates (in the form of error bars) for the time values plotted on the graph in the “analysis” section. Part 2 Now, use the brass cylinder (brass pendulum mass, minus the hanging hook screw) and repeat the data collection as in Part 1. Compute the averages and standard deviations for the speeds and elapsed times as done in Part 1.
Required Analysis
Make a full page graph (or two separate graphs if you prefer) of the speed on the vertical axis as a function of time on the horizontal axis for the wooden and brass cylinders, respectively. (Use two different plot symbols or colors to distinguish the data points for the wooden cylinder from those of the brass cylinder if you plot them on the same graph.) You may assume that the “clock” begins when the cylinder breaks the beam at the upper (first) photo gate. In other words, plot the speed at that point as occurring at zero on the time scale, which is the horizontal axis. Now plot the speeds of the cylinder at the lower photogates using the corresponding times of fall (elapsed times) from the upper gate to the respective lower photogates as the time values on the horizontal axis. Plot horizontal and vertical error bars based on your calculated standard deviations for each point plotted on the graph. In effect each point other than the one at time equal zero will be situated in the middle of an “error rectangle” since there is uncertainty in both the speed and the time. When you have finished the graph(s), address the question of how the speed changes with time. First draw separate best-fit smooth “curves” through your data points on the graphs for the wooden cylinder and the brass cylinder. Do the graphical results suggest a simple relationship between the speed and time? Explain. What can you say about the acceleration of the wooden cylinder and the brass cylinder? (Remember that acceleration is the change in speed divided by the corresponding time interval.) Be specific and quantitative. Don’t hesitate to query your lab instructor for ideas if you are in the “dark.”
Conclusion
Discuss what you have discovered about objects in so-called “free-fall.” What did you expect to find? Did your experiment agree with your expectations? Be quantitative. Discuss/explain. What factors may contribute to the “non-idealness” of the experiment?