Physics 0 esson 4 Simple Harmonic Motion Pendulums Refer to Chapter 7 in Pearson for a discussion of simple harmonic motion. I. Simple Harmonic Motion A study of simple harmonic motion (SHM) will take us from uniform circular motion and lead us into a study of Mechanical Waves. Simple pendulums and vibratin weihted sprins both exhibit SHM since they vibrate, swin, oscillate, or cycle with a reular pattern. An object is underoin SHM if both of the followin are true: An object or particle is vibratin with a constant frequency or period of motion. here is a restorin force toward a central equilibrium point. Recall from esson 9 (uniform circular motion) that Period () is the time to complete one swin, vibration, bounce, cycle or oscillation and that Frequency (f) is the number of swins, vibrations, bounces, cycles or oscillations in one second. Example What is the period and frequency of a vibratin sprin that requires 4 minutes and 40 seconds to complete 500 complete oscillations? totaltime number of vibrations f 80s f 500 0.56s = 0.56 s f.79hz II. he Simple Pendulum One day, many years ao, Galileo watched a candle chandelier (a simple pendulum) swinin back and forth on a lon chain in a cathedral. Galileo started to wonder what determined the period of the pendulum. What is the effect of chanin the lenth of the chain? If the mass of the chandelier was chaned, would that affect the period of vibration? And, if the amplitude (size of swin) was increased or decreased, would the period be affected? o answer these questions Galileo did an experiment similar to Activity. You should do Activity before proceedin with the remainder of the lesson. R.. & A.K. 4 7/0/7
III. Period of a Pendulum Equation Galileo found that the period of a pendulum s motion depends on the lenth of the pendulum, but it does not depend on the mass or amplitude (i.e. the maximum displacement of the swin). he period of a pendulum can be calculated usin: Where: period (s) lenth of pendulum (m) acceleration of ravity (m/s ) Example If a pendulum is 80.0 cm lon, what is its period and frequency of vibration? =.79 s 0.800m 9.8 m s f f f.79s 0.557Hz Example 3 On the planet called rathelmadore, Billy found that a 50.0 cm pendulum completed 0 swins in 33.6 s. What is the acceleration of ravity on rathelmadore? time for 0swins 0 33.6s 0.68s 4 4 (.68 s) 6.99 m 4 (0.500m) s Since is inside the radical, square everythin and then rearrane the equation for. R.. & A.K. 4 7/0/7
IV. Practice Problems. What is the period and frequency for a 0.75 m pendulum? (.74 s, 0.576 Hz). A pendulum requires 56.74 s to swin 0 times. What is the period, frequency, and lenth of the pendulum? (.84 s, 0.35 Hz,.0 m) 3. On the moon, a simple experiment was done to measure the acceleration of ravity. he period for a.0 m lon pendulum was measured and found to be 4.94 s. What is the acceleration due to ravity on the moon? (.6 m/s ) V. Hand-in Assinment. Explain what effect chanin each of the followin factors has on the period of a pendulum: (a) amplitude (b) ravitational field strenth (c) mass. A pendulum swins 400 times in 8 minutes and 0 seconds. What is its frequency and period of vibration? (0.80 Hz,.5 s) 3. If a pendulum is 35 cm lon: A. What is the period of the pendulum? (.33 s) B. How many swins will it complete in.5 minutes? (64) 4. If a pendulum takes 4.0 minutes to swin 400 times, what is the lenth of the pendulum? (0.089 m) 5. What is the acceleration due to ravity on a planet where an 80 cm pendulum requires 6.5 s to swin 50 times? (3.0 m/s ) 6. A pendulum swins with a period of 5.00 s on the Moon, where the ravitational field strenth is.6 m/s. What is the pendulum's lenth? (.03 m) *7. An inquisitive student brins a pendulum aboard a jet plane. he plane is in level fliht at an altitude of.3 km. What period do you expect for a 0.0 cm pendulum? (Hint: First determine the ravitational field strenth.) (0.898 s) R.. & A.K. 4 3 7/0/7
Activity - Period of a Pendulum Objective: Which of the followin factors effect the period of a pendulum? lenth of the pendulum mass of the pendulum bob the size of the swin or amplitude Observations: chane in mass mass lenth amplitude time for 0 full swins (s) 0 50 cm 0 cm 50 50 cm 0 cm 00 50 cm 0 cm chane in lenth mass lenth amplitude time for 0 full swins (s) any 0 cm 0 cm any 50 cm 0 cm any 90 cm 0 cm chane in amplitude mass lenth amplitude time for 0 full swins (s) any 50 cm 0 cm any 50 cm 0 cm any 50 cm 30 cm period (s) period (s) period (s) Conclusions: Based on your results, what factor(s) determine the period of a pendulum? Write-up Observation data tables like those above on a separate sheet of paper. Answer to the conclusion question. R.. & A.K. 4 4 7/0/7
Activity Measurin A. Measurin Usin the Free Fall Method An apparently simple way to measure the acceleration of ravity is to measure how lon it takes for an object to fall a known distance and then calculate the acceleration usin dv t a. t Materials: stop watch timer measurin tape rubber stopper Procedure:. In a suitable location (i.e. the south stairwell at the end of the science hallway) measure the time it takes for a rubber stopper to fall a measured distance. Measure the time three or four times and then choose the time that you feel was the most accurate measurement. Analysis:. Calculate the acceleration due to ravity based on your measurements. Do not "fix" your results to coincide with the actual value of the acceleration due to ravity. No marks are awarded or lost for a "poor" experimental value.. he actual value for the acceleration due to ravity is 9.8 m/s. Determine the percent error for your value usin the formula below: experimental value - actual value % error 00% actual value 3. In a sentence or two, evaluate this method for measurin the acceleration due to ravity. Write-up Neatly oranised data table of measurements and observations. Calculation of acceleration due to ravity and calculation of percent error. Evaluation of method. R.. & A.K. 4 5 7/0/7
B. Measurin Usin a Pendulum Objective: o determine the value of the acceleration due to ravity () usin a simple pendulum. heory: he period () of a pendulum s motion depends only on the lenth of the pendulum () and the acceleration due to ravity (). Manipulatin the equation we et: 4 rise slope run We can easily measure both the lenth () and the period of vibration () for a number of pendulum lenths. We can then calculate 4 and for each lenth of pendulum and then use these to plot a raph of 4 Vs. he slope of the best-fit-line should yield a very accurate value for, assumin that the measurements were carefully done. Procedure:. Usin the materials provided and a table like the one below, measure the period of vibration for different lenths (0.40 m, 0.50 m, 0.60 m, 0.70 m, 0.80 m, 0.90 m,.0 m) of the pendulum. (Accurately measurin the time for one swin is difficult. It is easier to measure the time for 0 or 0 swins and then divide by the number of swins to find the period.) (Be sure to measure the lenth from the top of the strin to the center of the mass.) enth time for 0 full (s) (s ) 4 (m) swins (s). Plot a raph of 4 Vs, draw a line of best fit and calculate the acceleration due to ravity. 3. How does your value compare with the accepted value of 9.8 m/s? What is the percent error? 4. Comment on the accuracy of your result. 5. Compare and contrast the two methods (free fall, pendulum) for measurin. Write-up Neatly oranised data table of measurements and observations. Neatly done raph and calculation of slope. Calculation of acceleration due to ravity and calculation of percent error. Comment on accuracy and comparison of methods. R.. & A.K. 4 6 7/0/7