Proving the Work Energy Theorem

Empirical Tests of Loop-the-Loop Physics 2A

Melvin J. Vaughn, Instructor

Written by Dave G

Experiment Date: April 24, 2011

Report Date: May 8, 2011

Proving the Work Energy Theorem

Empirical Tests of Loop-the-Loop

Contents

1.  Abstract............................................................ 3

2. Principles.......................................................... 6

3.  Experimental Procedure........................... 9

4.  Raw Data and Numerical Analysis...... 19

5.  Results............................................................ 24

6.  Discussion and Conclusion..................... 27 # 1.  Abstract

This lab report documents an experiment to empirically answer a quiz problem about movement of a pendulum. The problem to solve is:

A pendulum is formed with a small ball and string of length L. A peg is at height R above the pendulum's lowest point.

The question is, at what height H above the pendulum's lowest point must the pendulum weight must be released for the pendulum weight to swing completely around the peg, without the pendulum cord going slack.

This picture illustrates the question: The result of this experiment is that expected, experimental, and actual results are consistent within just over four percentage points. Calculations based on the work energy, empirical test results from this experiment, and generally accepted values found in textbooks, are consistent within just over four percent:

\E1     Expected result:

Using work energy principles the computed result is the minimum height H above the pendulum's lowest point the pendulum must be released in order to circle the peg without the cord going slack, is 2.5 times the value of radius R.

\E1     Experimental result:

The experimental result is the minimum height H above the pendulum's lowest point the pendulum must be released in order to circle the peg without the cord going slack, is 2.4 times the value of radius R.

\E1     Actual result:

Physics text books describing this and similar loop-the-loop problems consistently show a height H equal to 2.5 times loop radius R will allow the pendulum to circle the peg without the pendulum cord going slack.

# 2. Principles

There are three principles pertinent to this experiment:

1.  Conservation of energy

2.  Work energy theorem

3.  Relationship between angular acceleration and weight of a body at apex of vertical rotation

1.  Conservation of Energy

The principles of energy conservation state that total energy in a system remains constant. Energy can be transferred from one object to another but total energy is always the same. Work done by an applied force that changes elevation, of an object such as a pendulum in this case, is related to the work done by gravity and change in the object's kinetic and potential energy.

2.  Work Energy Theorem

The work energy theorem is used to prove conservation of energy. Kinetic and potential energies in a system before and after an event, such as linear or circular travel, sum to the same value.

The quiz problem was answered using work energy equations. Work energy can be used to estimate velocity of an object at a given elevation, elevations needed to achieve a certain velocities, and changes in kinetic and potential energy after linear or rotational movement.

The work energy theorem is represented with the following equation. It is used in this experiment to compute the expected value of height H:

Energy before  =  Energy after

mgho + 1/2 mvo^2  =  mgh + 1/2 mv^2

3. Angular acceleration is at least equal to weight at the apex of rotation.

The problem to solve in this experiment is similar to various problems seen in physics books commonly termed \D2loop the loop\D3 problems. In these problems, an object travels a circle perpendicular to the ground. At the apex of the circle, the object must have a certain angular acceleration in order to complete the circle. That acceleration must equal or exceed weight (mg) of the object. If this condition is not true, the object will not successfully complete the loop.

\E1     If the object is a roller coaster car on a track, normal force of the track on the car and weight both point down. If the normal force is zero, meaning the track is not pushing on the car, then angular acceleration is equal to weight. For the car to complete the loop, angular acceleration must equal or exceed weight.

The equation representing this principle is mv^2/r >= mg

\E1     If the object is a pendulum on a cord, then cord tension must equal or exceed weight (mg). For pendulum cord tension to equal or exceed weight, angular acceleration must equal or exceed weight.

The equation representing this principle, in the case of a pendulum on a cord, is mv^2/r >= mg.

# 3.  Experimental Procedure

This section describes what was done to complete the experiment. There are five steps in the experimental procedure:

1.  The first step in this experimental procedure is compute the answer to the quiz question using work energy.

2.  The second step in this experimental procedure is read and research the original quiz question, and similar problems, in various different physics text books, and determine whether there is a generally accepted solution to loop-the-loop problems.

3.  The third step this experimental procedure is build a test fixture.

4.  The fourth step in this experimental procedure is run tests to experimentally determine the answer to the quiz problem.

5.  The fifth step in this experimental procedure is analysis and interpretation of the results.

Details of each step in this experimental procedure are as follows:

Step 1 – Compute the value of H using work energy

The first step in this experimental procedure is answer the quiz question using work energy. The equation that is used to do this is:

1/2 mvo^2 + mgho = 1/2 mv^2 + mgh

where:

\E1     initial velocity vo of the pendulum is 0

\E1     h = 2r meaning final height H of the pendulum, at the apex of rotation when the cord is taught, is twice the value of R

\E1     cord tension at apex of the loop around the peg, to complete a loop around the peg, without the cord going slack, is equal to or greater than pendulum weight. This means:

o  mg = mv^2/r at h

o  v^2 = gr at h

o  v=sqrt(gr) at h

The original equation can then be written and simplified as:

mgho = 1/2 mv^2 + mg2r         (h = 2r)

mgho = 1/2mgr + 2mgr             (v^2 = gr)

ho = 1/2r + 2r = 2.5 r = H          (cancel m and g from all 3 terms)

The result of this step in the experimental procedure is an expected value of height H, and is shown in the computational procedure steps to be 2.5 times radius R of the circle described by the pendulum circling the peg.

Step 2 - Reading and research

In this step of the experimental procedure various physics textbooks are read and reviewed with the objective of finding problems similar to the original quiz problem. The books will show different perspectives on the same types of loop-the-loop problem. There is consistency in use of work energy and a conclusion that height H is 2.5 times radius R.

Consult the following textbooks:

\E1     Applied Physics, by Dale Ewen, Neill Schurter, and Erik Gundersen. Eighth edition. Pages 185-211.

\E1     Fundamentals of Physics, by Jearl Walker. Part 1, eighth edition. Pages 140-201.

\E1     Physics, by Robert Resnick, David Halliday, and Kenneth Krane. Part 1, fourth edition, pages 131-199 and the accompanying Selected Solutions guide. See also part 1, seventh edition of this same book, pages 140-200.

\E1     Physics, by James S. Walker. Volume 1, fourth edition, pages 216-253.

In each of these books, read about similar loop-the-loop problems, how they are solved using work energy, and the solution to the problem.

The result of this step is values for H and R that are used as accepted results, to be compared with expected and experimental results, and used to compute percent difference and percent error.

Step 3 - Construct Fixture

The third step in this experimental procedure is construct a test fixture. Follow these steps:

\E1     On a 17 x 40 inch piece of plywood, using a pen, draw a horizontal line along the short side two inches from the edge.

\E1     Mark the center of this line. Impale the center with a four-inch lag screw. This is the pendulum\D5s axis of rotation.

\E1     Mark one-inch intervals below the lag screw for 24 inches. Drill a 2 cm hole precisely at the one-inch intervals. Insert a nail that is 3 inches long and 2 cm in diameter in the first hole.

\E1     Inscribe and mark one-inch intervals below the horizontal line along the left side of the board.

\E1     Inscribe and mark degrees of an inverted semicircle, from a position equivalent to zero degrees on a unit circle, through 270 degrees, and up to 180 degrees, using the axis of rotation as the center.

\E1     Hang a 20.5 inch pendulum cord of 8lb test nylon and a 0.125 ounce lead weight on the end of the four-inch lag screw impaled at the horizontal center of the board two inches below the top.

\E1     Attach the board to a wall so it stands perpendicular to the floor.

Construction of the test fixture is complete.

Description of the finished product:

The test fixture is essentially an inverted protractor inscribed on a large sheet of plywood, with a pendulum suspended from the center of the semicircle formed by the protractor. Degrees of a semicircle, from 90 to 270 degrees, or from the positive to negative X axis via the negative Y axis, are marked on the plywood board at ten degree intervals. A scale at one-inch intervals is inscribed directly under the pendulum axis and along the left side of the board. Directly below the lag screw at one inch intervals are holes where a steel peg is inserted.  The peg is 2 centimeters in diameter

A photograph and a diagram of the test fixture are on the following two pages.

Step 4 - Find the experimental value of R

Having built the test fixture, the next step in this experimental test procedure is run a sequence of tests to answer the original quiz question and experimentally determine the value of R.

Fourteen tests are run, described as follows:

\E1     For each test, the pendulum is released at a position parallel to the axis of rotation with the cord fully extended. This position corresponds to 90 degrees on a unit circle. This release position is along the positive X axis, with the cord extended, and level with the pendulum axis of rotation.

\E1     The pendulum is first released with the peg in a hole one inch below the axis of rotation, meaning that L is R plus one inch. The pendulum swing is carefully observed for whether the pendulum circles the peg with the cord taught. The final position of the pendulum after a full swing is recorded and can be seen in the recorded test results.

\E1     With each subsequent test, the peg is moved down one inch and the pendulum was released again. Each time the peg is moved down, R decreases in length by one inch. The farthest location of the swing is recorded. The pendulum is carefully observed to see whether it circles the peg with the cord taught. The result of each test is recorded.

\E1     After the first time the pendulum swings a full circle around the peg with the cord taught, three more tests are conducted.

1.  The first test is to simply repeat the test, releasing the pendulum from a level that results in a full circle around the peg with the cord taught. The purpose is to verify the finding.

2.  The second test is to raise the peg one half an inch and then observe the swing. The purpose is to more precisely determine the value of R.

3.  The third test is to lower the peg to 13 inches below the axis, meaning R is now 7.5 inches, and test again. The purpose is to observe and record what happens. The interesting result of this test is recorded.

Step 5 - Analysis of the numerical results

At this point in the experimental procedure the experimental value of height H has been established. In this step of the experimental procedure, the relationship between experimental values of radius R and height H are computed. The experimental, expected, and actual values of R and H are compared. Percent error and percent difference are computed.

The experimental process is reviewed for measurement errors in order to estimate a margin of error. The process is also reviewed to look for areas of possible improvement, unanswered questions, and new potential areas of investigation.

This completes the experimental test procedure.

# 4.  Raw Data and Numerical Analysis

This section presents what was learned reading about loop the loop problems in textbooks, the computation of height H using work energy, and data from tests to empirically determine height H. The three sets of results are:

1.  The first result is the accepted value of height H, ascertained from review of physics books.

2.  The second result is the expected value of height H, computed using work energy theorem

3.  The third result is the value of height H, found by testing

1.  Accepted value of H ascertained form a literature search

The accepted value of height H, as gleaned from reading various physics texts that describe loop-the-loop problems, is 2.5R.

2.  Expected value of H computed using work energy

The minimum value for height H, of the pendulum, required to complete a full circle around the peg, is 2.5 times the value or radius R, above the lowest point of the pendulum's path. This result is computed using work energy. The computation is premised on the principle that tension on the pendulum cord at the apex of circular travel around the peg must equal or exceed pendulum weight.

The calculation of height H is as follows:

1/2 mvo^2 + mgho = 1/2 mv^2 + mgh

where:

\E1     initial velocity vo of the pendulum is 0

\E1     h = 2r meaning final height H of the pendulum is twice the value of radius R

\E1     cord tension at apex of loop around the peg, to complete a loop around the peg, without the cord going slack, is equal to or greater than pendulum weight. This means:

o  mg = mv^2/r at H, and            (cancel m and isolate v^2)

o  v^2 = gr at H, and

o  v=sqrt(gr) at H

The equation is rewritten and simplified as:

\E1     mgho = 1/2 mv^2 + mg2r       (h = 2r)

\E1     mgho = 1/2mgr + 2mgr           (v^2 = gr)

\E1     ho = 1/2r + 2r = 2.5 r = H        (cancel m and g from all 3 terms)

This means minimum value of height H is 2.5 times radius of the loop formed when the pendulum cord circles the peg.

3.   Experimental value of H derived from testing

The recorded test results show a radius R value of 8.5 inches leads to the pendulum completing a rotation around the peg with the cord remaining taught.

A nine-inch radius was tested. The test failed, meaning R has a margin of error of half an inch. There are three other sources of error. These four sources of error sum to 1.5 inches. Details of this finding are in the summary and conclusion.

The value of height H is 20.5 inches. Radius R for a successful rotation is 8.5 inches. H divided by R is 2.4. H is equal to 2.4R.

# 5.  Results

Test results prove the expected and actual result within just over four percentage points. When radius R is 8.5 inches, meaning 2.4R = H, the pendulum makes a full circle around the peg with the pendulum cord remaining taught. The experimental result is 2.4R, the expected result is 2.5R, and the actual result is 2.5R.

The result of this experiment is that calculations based on the work energy are consistent with empirical results:

\E1     The actual result from a literature search is H=2.5R.

\E1     The experimental result based on tests documented in this report is H=2.4R. The minimum height H above the pendulum's lowest point that the pendulum must be released in order for the pendulum to make it completely around the peg without going slack is equal to 2.4 times the value of R.

\E1     The expected result using work energy principles is H=2.5R. This computed result is that the minimum height H above the pendulum's lowest point that the pendulum must be released in order for the pendulum to make it completely around the peg without going slack is equal to 2.5 times the value of R.

Percent error and percent difference

\E1     Percent error is actual less measured over actual. The actual value of H=2.5R is taken from the literature search. The measured value of H=2.4R is the experimental result to two significant figures.

Percent error = abs(2.5-2.4)/2.5 = 4.0 percent

\E1     Percent difference is measured minus expected over the average of expected and measured. The measured value of H is 2.4R from the tests. The expected value of H is 2.5R, computed in this experiment using work energy.

Percent difference = abs(2.4-2.5)/((2.4+2.5)/2) = 4.1 percent.

# 6.  Discussion and Conclusion

Test Results

The results of this experiment support principles of energy conservation and use of work energy computations to prove energy conservation. The results show that work energy theorem equations accurately predict an empirically provable result. Work energy can be used to answer the original problem.

Experimental test results and computed results agree with accepted values. Physics textbooks with this kind of loop-the-loop problem use work energy theorem as the solution tool, and show a result of H = 2.5R.

Sources of Error

The experimental test result has sources of error relating to precision of measurement:

\E1     Calibrating degrees of a circle on to the plywood board that is the test fixture. It is estimated the degree values could be off by +/- 2 degrees

\E1     Calibrating the exact number of inches below the pendulum axis where holes are drilled for the peg. It is estimated the measurements could be off by +/- 1/8 of an inch.

\E1     Reading inches below the plane of the axis of rotation the pendulum swings, as recorded in tests 1 to 8. It is estimated this reading could be off by +/- half an inch.

\E1     Air resistance that the pendulum is exposed to. Given the minimal size of the pendulum cord and weight it is expected this error is very small.

\E1     The recorded test notes show that a radius R value of 8.5 inches leads to the pendulum completing a circle around the peg with the cord remaining taught. Nine inches was tested and the test failed, suggesting the R value has a half-inch margin of error.

\E1     Measuring the pendulum cord length is estimated to be accurate to within an eighth of an inch.

\E1     Consistently releasing the pendulum from the same release point is estimated to vary by as much as one fourth of an inch.

These possible errors in measurement sum to an inch and a half.

Personal observations

The original question appears to be a variation of a common question I have seen in a variety of different physics textbooks. It seems to be something of a classic question. It is commonly called a 'loop the loop' question. The same problem is illustrated using a roller coaster that must safely run a full circle without separating from its tracks.

A number of interesting test results were seen during the design, execution, and analysis in this experiment. Unfortunately there was not time to explore all the issues that were encountered. Expanding the scope of investigation and refining measurement techniques could improve the experiment.

How can this experiment be improved?

\E1     This experiment could be improved by developing a more carefully calibrated test fixture. There were precision errors in marking degrees of the angles, depth below the axis, and height of swing. Using a smooth board, more precise measurement tools, and more careful inscribing would improve precision.

\E1     Executing the test in a vacuum would eliminate air resistance, which was not considered in calculations.

\E1     Capturing the pendulum swings on video may allow more precise determination of exactly how high it swings.

\E1     Exploring and answering some of the unanswered questions, listed below, will help improve the informational and educational value of this experiment.

This experiment raises more questions answers. Many interesting observations were made that are not explained or understood. Further testing, investigation, and help from other teachers and students could help improve the experiment by covering more areas of investigation.

Some of the unanswered questions raised by this experiment are:

\E1     With test swings when the pendulum remained taught but did not rotate, how does the observed height compare to a computed estimate and how would that computation best be done? See tests #1-8.

\E1     What further information can be determined from the two tests where the pendulum rotated a full circle but did not keep the pendulum cord taught? See tests #9-11

\E1     What is the explanation for the final test, where R was reduced to 7.5 inches? The pendulum cord completely wound around the peg and appeared to still have momentum. Why did this happen? See test #13.

\E1     If the cord did not make it around the loop, it did not go slack. What is the explanation for this result? The expected result is the cord goes slack. See tests #9-11.

\E1     When the pendulum first made a circle, the cord was slack. If the cord was slack, a circle was completed. The cord never went slack on a failed attempt to complete a circle. Why this difference in observed behavior? See tests #1-8.

\E1     The ratio of pendulum velocity at apex and lowest point is of the circle is an interesting constant value. I would like to better understand this mathematical relationship and the reasons for it. -- end of document --