by Dr. James E. Parks Department of Physics and Astronomy 401 Niesen Physics Buidin The University of Tennessee Knoxvie, Tennessee 37996-100 Copyriht June, 000 by James Edar Parks* *A rihts are reserved. No part of this pubication may be reproduced or transmitted in any form or by any means, eectronic or mechanica, incudin photocopy, recordin, or any information storae or retrieva Objectives The purposes of this experiment are: (1) to study the motion of a simpe penduum, () to study simpe harmonic motion, (3) to earn the definitions of period, frequency, and ampitude, (4) to earn the reationships between the period, frequency, ampitude and enth of a simpe penduum and (5) to determine the acceeration due to ravity usin the theory, resuts, and anaysis of this experiment. Theory A simpe penduum may be described ideay as a point mass suspended by a massess strin from some point about which it is aowed to swin back and forth in a pace. A simpe penduum can be approximated by a sma meta sphere which has a sma radius and a are mass when compared reativey to the enth and mass of the iht strin from which it is suspended. If a penduum is set in motion so that is swins back and forth, its motion wi be periodic. The time that it takes to make one compete osciation is defined as the period T. Another usefu quantity used to describe periodic motion is the frequency of osciation. The frequency f of the osciations is the number of osciations that occur per unit time and is the inverse of the period, f = 1/T. Simiary, the period is the inverse of the frequency, T = /f. The maximum distance that the mass is dispaced from its equiibrium position is defined as the ampitude of the osciation. When a simpe penduum is dispaced from its equiibrium position, there wi be a restorin force that moves the penduum back towards its equiibrium position. As the motion of the penduum carries it past the equiibrium position, the restorin force chanes its direction so that it is sti directed towards the equiibrium position. If the restorin force F is opposite and directy proportiona to the dispacement x from the equiibrium position, so that it satisfies the reationship
F = - k x (1) then the motion of the penduum wi be simpe harmonic motion and its period can be cacuated usin the equation for the period of simpe harmonic motion T = π m k. () It can be shown that if the ampitude of the motion is kept sma, Equation () wi be satisfied and the motion of a simpe penduum wi be simpe harmonic motion, and Equation () can be used. T T x F m F = m sin m a b c Fiure 1. Diaram iustratin the restorin force for a simpe penduum. The restorin force for a simpe penduum is suppied by the vector sum of the ravitationa force on the mass. m, and the tension in the strin, T. The manitude of the restorin force depends on the ravitationa force and the dispacement of the mass from the equiibrium position. Consider Fiure 1 where a mass m is suspended by a strin of enth and is dispaced from its equiibrium position by an ane and a distance x aon the arc throuh which the mass moves. The ravitationa force can be resoved into two components, one aon the radia direction, away from the point of suspension, and one aon the arc in the direction that the mass moves. The component of the ravitationa force aon the arc provides the restorin force F and is iven by F = - m sin (3) Revised 10/5/000
where is the acceeration of ravity, is the ane the penduum is dispaced, and the minus sin indicates that the force is opposite to the dispacement. For sma ampitudes where is sma, sin can be approximated by measured in radians so that Equation (3) can be written as F = - m. (4) The ane in radians is x, the arc enth divided by the enth of the penduum or the radius of the circe in which the mass moves. The restorin force is then iven by x F = - m (5) and is directy proportiona to the dispacement x and is in the form of Equation (1) where m k =. Substitutin this vaue of k into Equation (), the period of a simpe penduum can be found by and T = π m (6) m ( ) T = π. (7) Therefore, for sma ampitudes the period of a simpe penduum depends ony on its enth and the vaue of the acceeration due to ravity. Apparatus The apparatus for this experiment consists of a support stand with a strin camp, a sma spherica ba with a 15 cm enth of iht strin, a meter stick, a vernier caiper, and a timer. The apparatus is shown in Fiure. Procedure 1. The simpe penduum is composed of a sma spherica ba suspended by a on, iht strin which is attached to a support stand by a strin camp. The strin shoud be approximatey 15 cm on and shoud be camped by the strin camp between the two fat pieces of meta so that the strin aways pivots about the same point. Revised 10/5/000 3
Fiure. Apparatus for simpe penduum.. Use a vernier caiper to measure the diameter d of the spherica ba and from this cacuate its radius r. Record the vaues of the diameter and radius in meters. 3. Prepare an Exce spreadsheet ike the exampe shown in Fiure 3. Adjust the enth of the penduum to about.10 m. The enth of the simpe penduum is the distance from the point of suspension to the center of the ba. Measure the enth of the strin s from the point of suspension to the top of the ba usin a meter stick. Make the foowin tabe and record this vaue for the enth of the strin. Add the radius of the ba to the strin enth s and record that vaue as the enth of the penduum = s + r. 4. Dispace the penduum about 5º from its equiibrium position and et it swin back and forth. Measure the tota time that it takes to make 50 compete osciations. Record that time in your spreadsheet. 5. Increase the enth of the penduum by about 0.10 m and repeat the measurements made in the previous steps unti the enth increases to approximatey 1.0 m. 6. Cacuate the period of the osciations for each enth by dividin the tota time by the number of osciations, 50. Record the vaues in the appropriate coumn of your data tabe. Revised 10/5/000 4
Fiure 3. Exampe of Exce spreadsheet for recordin and anayzin data. 7. Graph the period of the penduum as a function of its enth usin the chart feature of Exce. The enth of the penduum is the independent variabe and shoud be potted on the horizonta axis or abscissa (x axis). The period is the dependent variabe and shoud be potted on the vertica axis or ordinate (y axis). 8. Use the trendine feature to draw a smooth curve that best fits your data. To do this, from the main menu, choose Chart and then Add Trendine... from the dropdown menu. This wi brin up a Add Trendine diao window. From the Trend tab, choose Power from the Trend/Reression type seections. Then cick on the Options tab and seect Dispay equations on chart option. 9. Examine the power function equation that is associated with the trendine. Does it suest the reationship between period and enth iven by Equation (7)? 10. Examine your raph and notice that the chane in the period per unit enth, the sope of the curve, decreases as the enth increases. This indicates that the period increases with the enth at a rate ess than a inear rate. The theory and Equation (7) predict that the period depends on the square root of the enth. If both sides of Equation 7 are squared then Revised 10/5/000 5
4π T=. (8) If the theory is correct, a raph of T versus shoud resut in a straiht ine. 11. Square the vaues of the period measured for each enth of the penduum and record your resuts in the spreadsheet. 1. Use the chart feature aain to raph the period squared, T, as a function of the enth of the penduum. The period squared is the dependent variabe and shoud be potted on the y axis. The enth is the independent variabe and shoud be potted on the x axis. 13. Examine your raph of T versus and check to see if there is a inear reationship between T and so that the data points ie aon a ine. 14. Use the trendine feature to perform a inear reression to find a straiht ine that best fits your data points. This time from the Add Trendine diao window. choose Linear from the Trend/Reression type seections. Cick on the Options tab and once aain seect the Dispay equations on chart option. This shoud draw a straiht ine that best fits the data and shoud dispay the equation for this straiht ine. 15. Equation (8), 4π T=, is of the form y=ax+b where y=t, a = 4π, x=, and b=0. A raph of T versus shoud therefore resut in a straiht ine whose sope, a, is 4π equa to. From the equation for the trendine, record the vaue for the sope, a, and from the equation a= 4π find, the acceeration due to ravity. 16. Compare your resut with the accepted vaue of the acceeration due to ravity 9.8 m/s. Cacuate the percent difference in your resut and the accepted resut. % Difference = [(your resut - accepted vaue)/accepted vaue] x 100% 17. Usin the accepted vaue of the acceeration due to ravity and Equation 7 cacuate the period of a simpe penduum whose enth is equa to the onest enth measured in Tabe 1. Compare this theoretica resut with the measured experimenta resut and cacuate the percent difference. % Difference = [(Experimenta Resut - Theoretica Resut)/Theoretica Resut] x 100%. 18. The equation for the period of a simpe period, Equation (7), was deveoped by assumin that the ampitude is sma. The rane of ampitudes over which Equation Revised 10/5/000 6
(7) is vaid is to be determined by measurin the period of a simpe penduum with different ampitudes. 19. Adjust the enth of the penduum to about 0.6 m. Measure the period of the penduum when it is dispaced 5, 10, 15, 0, 5, 30, 40, 50, and 60 from its equiibrium position. Make a tabe to record the period T as a function of the ampitude A. 0. Usin your data, make a raph of the period versus the ampitude. 1. Measure the enth of the penduum and use Equation (7) to cacuate the period of the penduum. Add this theoretica point to your raph for the period with zero ampitude.. Examine your raph for the behavior of the period with ampitude. What concusions can you draw from your data reardin the rane of ampitudes over which Equation (7) is vaid? Questions 1. How woud the period of a simpe penduum be affected if it were ocated on the moon instead of the earth?. What effect woud the temperature have on the time kept by a penduum cock if the penduum rod increases in enth with an increase in temperature? 3. What kind of raph woud resut if the period T were raphed as a function of the square root of the enth,. 4. What effect does the mass of the ba have on the period of a simpe penduum? What woud be the effect of repacin the stee ba with a wooden ba, a ead ba, and a pin pon ba of the same size? Revised 10/5/000 7