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Topic: Max Tension in a Pendulum String (Read 8202 times)
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Posted on: 2002-05-09 11:09:00 by Edward P. Wyrembeck
Author
Edward P. Wyrembeck
Abstract
I have designed a simple experiment to test the theoretical prediction that the maximum tension in a pendulum string swinging down from a 90 degree angle from the equilibrium position is equal to 3 times the weight of the pendulum bob. I derive the prediction from well-known equations for nonuniform circular motion and conservation of mechanical energy in an isolated system. I use MBL tools to make the measurements and analyze the data. The experiment is appropriate for high school and college students.
Bridge Swinging and The Maximum Tension in a Pendulum String
While watching a Real Television clip during lunch I came across an interesting physics problem for high school and college students. It involved a number of thrill seekers that were bridge swinging. Bridge swinging is simply attaching a cable to a high bridge and then sitting on a small support at the end of that stretched out cable while you swing down and up on a simple pendulum. Unfortunately, for these thrill seekers they underestimated the maximum cable tension required to support and turn themselves as the pendulum bob and the cable broke. I didn.t know what the maximum string tension was for a pendulum bob swinging down from a 90-degree angle from the equilibrium position, so I did what any good physicist would do and I looked for some equations to guide my thinking. The equations1 I decided upon are well known and easy to use.
T = mv2/r + (mg) cos (q) (1)
Equation (1) relates the radial tension of a pendulum string to the sum of the inward radial centripetal force acting on the bob and the outward radial component of the weight of the bob (Fig. 1). Equation (1) predicts the tension in the pendulum string to be at maximum at the lowest position of the bob because the velocity of the bob will be at a maximum at the lowest position and the cos (q) will be at a maximum at 0 degrees the lowest position. All the other variables are constants.
(mg) h = (1/2) mv2 (2)
Equation (2) relates the gravitational potential energy of the bob on the bridge, with the zero of potential energy set at the lowest position of the bob (equilibrium position where it comes to rest), to the kinetic energy of the bob at the lowest position. This is the conservation of mechanical energy and by solving for v we can determine the theoretical maximum velocity of the bob at its lowest position, which turns out to be:
V = Ö2gh) (3)
Now we can substitute the Ö2gh) into equation (1) for the maximum velocity, where r = h, and arrive at the elegantly simple equation for the maximum tension of a pendulum string swinging down from q = 90 degrees.
MAX T = 3 (mg) (4)
Equation (4) tells us that the maximum tension in a simple pendulum string swinging down from a 90-degree angle is independent of the length of the string, depends only on the mass of the bob, and is 3 times the weight of the bob. This is a surprising result to most physics students and teachers alike and best of all one that can be tested with just a force sensor and a pulley.
We experimentally test equation (4) by attaching a good nylon pendulum string to a Vernier Dual-Range Force Sensor and pass the string over a PASCO Super Pulley, so the pulley acts as the pivot point for the pendulum. I use an old pool ball with an eyehook attached as the pendulum bob. To attach the eyehook I just drill a small diameter hole in the pool ball appropriate for the size of the eyehook and screw it in by hand. I get old pool balls for free from our local pool table supplier. We use a Vernier LabPro interface with LoggerPro software for pendulum string tension measurements and data analysis. Because this event is very brief, we use a high data-sampling rate of 5000 samples/s for a 1.0 s interval and trigger the force sensor automatically when the force exceeds 0.5 N.
To perform the experiment we simply attach the string to the force sensor set at the ± 10.0 N range, run it over the pulley, have a student hold the pool ball bob at the 90-degree angle (parallel with the floor) and let it go. Students measure the height of the string in the pulley and measure the height of the pool ball bob to make sure they are the same, so that q = 90 degrees. The force verses time graph will have some small amplitude vibrations imposed on the curve due to what we believe to be some slack in the pendulum string when the student releases the bob and vibrations in the pulley and support structure for the pulley.
In order to reduce this effect we mechanically release the bob by attaching a string from one end of the bob to the other with another eyehook to form a sling and attach another string to the middle of the sling to release the bob. We attach the vertical sling string to a pendulum clamp and release the bob smoothly by burning the string (Fig. 2). This reduces the amplitude vibrations on the force verses time graph noticeably (Fig. 3).
To compare the theoretical prediction of equation (4) MAX T = 3 (mg) to the measured maximum pendulum string tension, we first hang the bob over the pulley at q = 0 degrees and collect force data and save it as our reference bob weight. We use LoggerPro to create a new column of data (Max Tension) equal to 3 (mg) and save it. This gives the students a visual reference for the predicted maximum tension of the pendulum string that they can easily see and compare with the peek of the force verses time graph (Fig. 3). Students measure the maximum string tension from the graph by calculating the mean peak force over a short interval of time (0.05 s) centered on the peak force (Fig. 3). Students calculate the percent difference between the predicted and measured maximum string tension with typical results having less than a 2% difference. Table I is a student data table of their results. If time permits, students can confirm that the maximum tension in the string is independent of the length of the string by simply changing the length of the string and dropping the bob again while collecting data.
I believe this is an excellent experiment for high school and college physics students because the results are surprising and elegantly simple, it combines the fundamental physics concepts of centripetal force and conservation of mechanical energy in an isolated system, and it can easily be tested with widely used MBL tools. Also, students always remember, if you are going to go swinging, make sure the cable can withstand 3 times your weight or you may get another physics lesson in projectile motion.
A special thanks to my physics students: Josh, Nick, Josh, Ryan, and Cory. This note would not have been possible without their help and support.
Table 1. Student data table
Mass of Bob(kg) Weight of Bob (mg)(N) PredictedMax Tension(N) MeasuredMax Tension(N) PercentDifference
0.156 1.53 4.59 4.58 0.22%
References
1. P.M. Fishbane, S. Gasiorowicz, and S.T. Thorton, Physics for Scientists and Engineers, (Prentice-Hall, Inc., New Jersey, 1993), p. 147.
Edward P. Wyrembeck ph: 920-565-4450 (School)
Physics & Calculus Teacher ph: 920-458-4481 (Home)
Howards Grove High School fx: 920-565-4451
401 Audubon Road email: ewyrembe@hgsd.k12.wi.us (School)
Howards Grove, WI 53083 email: ewyrembeck@milwpc.com (Home)
web site: http://hghstigers.freehomepage.com/
web site: http://hghstigers.freehomepage.com/wyrembeckfamily/
The Fundamental Theorem: one exists for others!
http://www.aip.org/history/einstein/essay.htm
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AndyDerousselle
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Posts: 1
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That's very clever the way you've gone about working it out. I can't fathom if it's correct or not but I like the way you've gone about working it out.
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