Simple PendulumA simple pendulum is one which can be considered to be a point mass suspended from a string or rod of negligible mass. It is aresonantsystem with a single resonant frequency. For small amplitudes, theperiodof such a pendulum can be approximated by:Show

Top of FormForpendulum lengthL =cm =m

andacceleration of gravityg =m/s2

thependulum periodisT =s

Bottom of Form(Enter data for two of the variables and then click on the active text for the third variable to calculate it.)Note that the angular amplitude does not appear in the expression for the period. This expression for period is reasonably accurate for angles of a few degrees, but the treatment of thelarge amplitude pendulumis much more complex.If the rod is not of negligible mass, then it must be treated as aphysical pendulum.

Sean Carroll relates the story of Galileo's discovery of the fact that for small amplitudes, the period and frequency are unaffected by the amplitude. "In 1581, a young Galileo Galilei reportedly made a breakthrough discovery while he sat bored during a church service in Pisa. The chandelier overhead would swing gently back and forth, but it seemed to move more quickly when it was swinging widely (after a gust of wind, for example) and more slowly when it wasn't moving as far. Intrigued, Galileo decided to measure how much time it took for each swing, using the only approximately periodic event to which he had ready access: the beating of his own pulse. He found something interesting: The number of heartbeats between swings of the chandelier was roughly the same, regardless of whether the swings were wide or narrow. The size of the oscillations - how far the pendulum swung back and forth - didn't affect the frequency of those oscillations." Index

Periodic motion concepts

CarrollEternity to Here, p16

HyperPhysics*****MechanicsR Nave

Go Back

Pendulum MotionThe motion of asimple pendulumis likesimple harmonic motionin that the equation for the angular displacement isShow

which is the same form as the motion of a mass on a spring:

The anglular frequency of the motion is then given by

compared tofor a mass on a spring.

The frequency of the pendulum in Hz is given by

and the period of motion is then

.Index

Periodic motion concepts

HyperPhysics*****Mechanics R Nave

Go Back

Period of Simple PendulumA point mass hanging on a massless string is an idealized example of asimple pendulum. When displaced from itsequilibriumpoint, the restoring force which brings it back to the center is given by:Show

For small angles , we can use the approximationShow

in which caseNewton's 2nd lawtakes the form

Even in this approximate case, the solution of the equation uses calculus and differential equations. Thedifferential equationis

and for small angles the solution is:

Index

Periodic motion concepts

HyperPhysics*****MechanicsR Nave

Go Back

Pendulum GeometryThe period of asimple pendulumfor small amplitudes is dependent only on the pendulum length and gravity. For thephysical pendulumwith distributed mass, the distance from the point of support to the center of mass is the determining "length" and the period is affected by the distribution of mass as expressed in themoment of inertia I.

Index

Periodic motion concepts

HyperPhysics*****MechanicsR Nave

Go Back

Pendulum EquationThe equation of motion for thesimple pendulumfor sufficiently small amplitude has the formwhich when put inangular formbecomes

Thisdifferential equationis like that for thesimple harmonic oscillatorand has the solution:

Index

Periodic motion concepts

HyperPhysics*****MechanicsR Nave

Go Back

1996-2012 The Physics Classroom, All rights reserved.The Period of a PendulumA simple pendulum consists of a light string tied at one end to a pivot point and attached to a mass at the other end. The period of a pendulum is the time it takes the pendulum to make one full back-and-forth swing. A group of students are investigating the factors that might affect the period of a pendulum. They conduct three different controlled experiments.Experiment 1In the first experiment, the students make a pendulum by hanging a 200.0-gram mass on the end of a string. They pull the mass back such that it makes an angle of 30 with its usual vertical orientation. They then release the mass, allowing it to swing back and forth. They use a stopwatch to measure the time it takes the pendulum to complete five full swings. They use this time to determine the period. They vary the length of the string while keeping the mass and angle constant. A plot of their data is shown in Figure 1.Experiment 2The students make a pendulum by hanging masses on the end of a string to create a pendulum length of 65 cm. They release the mass from rest after pulling it back 30 from its vertical orientation. They conduct several trials using varying amounts of mass hanging on the end of the string. A plot of their data is shown in Figure 2.Experiment 3The students make a pendulum by hanging a 200.0-gram mass on the end of a string to create a pendulum length of 67 cm. They conduct several trials while varying the angle that the string makes with the vertical orientation. A plot of their data is shown in Figure 3. 1996-2012 The Physics Classroom, All rights reserved. 1996-2012 The Physics Classroom, All rights reserved.Questions:1. Which statement describes the purpose of this collection of three experiments?a. To vary several properties of a pendulum.b. To use a pendulum in order to determine its period.c. To investigate the effect of mass, angle and length upon a pendulum's period.d. To measure the time for five complete swings so as to calculate a pendulum's period.2. Based on the three experiments, what conclusion can be made about the factors that affect the period of a pendulum?a. The mass and angle are the only factors that affect the period of a pendulum.b. The mass, the angle and the length are the three variables affecting the period.c. Given the many variables, it is impossible to tell which one(s) affect the period.d. Mass and angle have little effect; the length is the main factor affecting the period.3. Why did the students measure the time for five complete swings?b. The period is the time it takes to make five complete swings.a. The experiment doesn't take much time and they didn't want to be idle.c. Since each experiment included five trials, they needed to be consistent by timing five complete swings.d. Calculating the period from the time for five swings reduces error by providing an average of five periods.4. What was the control variable(s) in Experiment 1?a. Length b. Periodc. Length and period d. Mass and angle5. Which statement describes the effect of increasing length upon the period of a pendulum?a. An increase in the length has no effect upon the period of a pendulum.b. An increase in the length causes the period of a pendulum to increase.c. An increase in the length causes the period of a pendulum to decrease.d. As length increases, the period of a pendulum first increases and later decreases.6. Use Figure 1 to estimate the period of a pendulum with a length of 90 cm and a mass of 200.0 grams that is released from an angle of 30.a. 1.88 seconds b. 2.00 secondsc. 2.14 seconds d. 2.90 seconds7. A 130-cm length pendulum consisting of a 200.0-gram mass is released from an angle of 30. Assume the relationship between length and period shown in Figure 1 can be extrapolated beyond a length of 100 cm. What is a reasonable estimate of this pendulum's period?a. 2.12 seconds b. 2.55 secondsc. 3.30 seconds d. 3.40 seconds 1996-2012 The Physics Classroom, All rights reserved.8. What was the independent variable(s) in Experiment 2?a. Mass b. Periodc. Mass and period d. Length and angle9. What would be the most appropriate conclusion statement for Experiment 2?a. A pendulum can be used to determine if mass affects the period.b. By systematically varying the mass, one can determine if its affects the period.c. Large variations in mass have little to no effect upon the period of the pendulum.d. If the period of a pendulum is changed, there will be no resulting change in its mass.10. Use Figure 2 to estimate the period of a pendulum with a length of 65 cm and a mass of 400.0 grams that is released from an angle of 30.a. 0.81 secondsb. 1.30 secondsc. 1.63 secondsd. Nonsense! It is impossible to make such an estimate since no pattern exists.11. What was the dependent variable(s) in Experiment 3?a. Angle b. Periodc. Angle and period d. Length and mass12. A pendulum consists of a 100-gram mass suspended on a string to make a 50.0-cm length pendulum. The pendulum is released from an angle of 20. Why is it or is it not possible to use the data provided in these experiments to estimate the period of such a pendulum?a. A trial was not performed with these three parameters; so it is not possible to estimate the period.b. There is no clear pattern as to how the period depends upon the mass and the angle; and so the period cannot be determined.c. Length is the only variable that has any appreciable effect upon the period; so the 50.0 cm length can be used to determine the period from Figure 1.d. Separate graphs are provided for each of the three values; so each value can be used to determine three separate periods and then the average value can be calculated.13. Consider the following values for mass, length and release angle for a simple pendulum. Which pendulum would have the largest period?a. mass = 200 grams; length = 100 cm; release angle = 20b. mass = 500 grams; length = 50 cm; release angle = 50c. mass = 1000 grams; length = 80 cm; release angle = 50d. mass = 1000 grams; length = 80 cm; release angle = 20

Resource LessonPeriod of a Pendulum

Printer Friendly Version

Asimple pendulumconsists of a string, cord, or wire that allows a suspended mass to swing back and forth. The categorization of "simple" comes from the fact that all of the mass of the pendulum is concentrated in its "bob" - or suspended mass.

As seen in this diagram, thelengthof the pendulum is measured from the pendulum's point of suspension to the center of mass ofits bob. Itsamplitudeis the string's angular displacement from its vertical or itsequilibrium position. If a pendulum is pulled to the right side and released to swing back and forth, its path traces our a sine curve as shown below.

The time required for one complete vibration, for example, from one crest to the next crest, is called the pendulum'speriodand is measured in seconds.The formula to calculate this quantity is

where Lis the length of the pendulum in meters gis thegravitational field strength, or acceleration due to gravityThis quantity at sea level is 9.81 m/sec2and can be calculated as

where G= 6.67 x 10-11nt m2/kg2 MEarthis the mass of the earth (6.02 x 1024kg) REarthis the average radius of the earth (6.4 x 106meters)Notice in the formula that the mass of a simple pendulum's bob does not affect the pendulum's period; it will however affect thetensionin the pendulum's string.In thisrelated lesson, you will find aderivationof this formula for the period of a simple pendulumthat will help youunderstand the restrictions on its use. It will also explain to you why a simple pendulum is NOT a true representation of simple harmonic motion,SHM. Take a few moments and use thisphysletto investigate how the period of a pendulum is impacted by its length and its initial displacement.Thefrequencyof a pendulum represents the number of vibrations per second. This quantity is measured in hertz (hz) and is the reciprocal of the pendulum's period.

Let's practice a few problems with these formulas.

What would be the period of a pendulum locatedat sea level if it is 1.5 meters long?

If the pendulum's length were to be shortened to one-fourth its original value, what would be its new period?

How many complete vibrations would this shorter pendulum trace outin one minute if it were to be released with a small initial amplitude?

At sea level, how longwould apendulum be ifit has a frequency of 2 hz?

The timing mechanism in a grandfather's clock is based on the principles of a simple pendulum. If your clock is gaining time, should you shorten or lengthen its pendulum?

Would a grandfather clock keep time on the moon?

Aphysical pendulumcould be illustrated by swinging a meter stick about one end or a baseball bat about one end. The formula to calculate the period of a physical pendulum is

where is the pendulum'smoment of inertiameasured in kg m2 mis its mass in kilograms gis the local gravitational field strength or acceleration due to gravity Lis the moment arm or perpendicular distance from the pivot point to object's center of mass measured in meters

Related Documents

PhysicsLABCopyright 1997-2014Catharine H. ColwellAll rights reserved.ApplicationProgrammerMark Acton

The Simple Pendulum by Dr. James E. Parks Department of Physics and Astronomy 401 Nielsen Physics Building The University of Tennessee Knoxville, Tennessee 37996-1200 Copyright June, 2000 by James Edgar Parks* *All rights are reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage or retrieval ObjectivesThe purposes of this experiment are: (1) to study the motion of a simple pendulum, (2) to study simple harmonic motion, (3) to learn the definitions of period, frequency, and amplitude, (4) to learn the relationships between the period, frequency, amplitude and length of a simple pendulum and (5) to determine the acceleration due to gravity using the theory, results, and analysis of this experiment. Theory A simple pendulum may be described ideally as a point mass suspended by a massless string from some point about which it is allowed to swing back and forth in a place. A simple pendulum can be approximated by a small metal sphere which has a small radius and a large mass when compared relatively to the length and mass of the light string from which it is suspended. If a pendulum is set in motion so that is swings back and forth, its motion will be periodic. The time that it takes to make one complete oscillation is defined as the period T. Another useful quantity used to describe periodic motion is the frequency of oscillation. The frequency f of the oscillations is the number of oscillations that occur per unit time and is the inverse of the period, f = 1/T. Similarly, the period is the inverse of the frequency, T = l/f. The maximum distance that the mass is displaced from its equilibrium position is defined as the amplitude of the oscillation. When a simple pendulum is displaced from its equilibrium position, there will be a restoring force that moves the pendulum back towards its equilibrium position. As the motion of the pendulum carries it past the equilibrium position, the restoring force changes its direction so that it is still directed towards the equilibrium position. If the restoring force FG is opposite and directly proportional to the displacement x from the equilibrium position, so that it satisfies the relationship The Simple Pendulum Revised 10/25/2000 2 F = - k x G G (1) then the motion of the pendulum will be simple harmonic motion and its period can be calculated using the equation for the period of simple harmonic motion m T = 2k . (2) It can be shown that if the amplitude of the motion is kept small, Equation (2) will be satisfied and the motion of a simple pendulum will be simple harmonic motion, and Equation (2) can be used. mg mgT TF = mg sin x Fa b clFigure 1. Diagram illustrating the restoring force for a simple pendulum. The restoring force for a simple pendulum is supplied by the vector sum of the gravitational force on the mass. mg, and the tension in the string, T. The magnitude of the restoring force depends on the gravitational force and the displacement of the mass from the equilibrium position. Consider Figure 1 where a mass m is suspended by a string of length l and is displaced from its equilibrium position by an angle and a distance x along the arc through which the mass moves. The gravitational force can be resolved into two components, one along the radial direction, away from the point of suspension, and one along the arc in the direction that the mass moves. The component of the gravitational force along the arc provides the restoring force F and is given by F = - mg sin (3) The Simple Pendulum Revised 10/25/2000 3where g is the acceleration of gravity, is the angle the pendulum is displaced, and the minus sign indicates that the force is opposite to the displacement. For small amplitudes where is small, sin can be approximated by measured in radians so that Equation (3) can be written as F = - mg . (4) The angle in radians is xl, the arc length divided by the length of the pendulum or the radius of the circle in which the mass moves. The restoring force is then given by x F = - mg l (5) and is directly proportional to the displacement x and is in the form of Equation (1) where mg k = l . Substituting this value of k into Equation (2), the period of a simple pendulum can be found by ( )m T = 2mgl (6) and T = 2gl . (7) Therefore, for small amplitudes the period of a simple pendulum depends only on its length and the value of the acceleration due to gravity. Apparatus The apparatus for this experiment consists of a support stand with a string clamp, a small spherical ball with a 125 cm length of light string, a meter stick, a vernier caliper, and a timer. The apparatus is shown in Figure 2. Procedure 1. The simple pendulum is composed of a small spherical ball suspended by a long, light string which is attached to a support stand by a string clamp. The string should be approximately 125 cm long and should be clamped by the string clamp between the two flat pieces of metal so that the string always pivots about the same point. The Simple Pendulum Revised 10/25/2000 4Figure 2. Apparatus for simple pendulum. 2. Use a vernier caliper to measure the diameter d of the spherical ball and from this calculate its radius r. Record the values of the diameter and radius in meters. 3. Prepare an Excel spreadsheet like the example shown in Figure 3. Adjust the length of the pendulum to about .10 m. The length of the simple pendulum is the distance from the point of suspension to the center of the ball. Measure the length of the string sl from the point of suspension to the top of the ball using a meter stick. Make the following table and record this value for the length of the string. Add the radius of the ball to the string length sl and record that value as the length of the pendulum s ll r = + . 4. Displace the pendulum about 5 from its equilibrium position and let it swing back and forth. Measure the total time that it takes to make 50 complete oscillations. Record that time in your spreadsheet. 5. Increase the length of the pendulum by about 0.10 m and repeat the measurements made in the previous steps until the length increases to approximately 1.0 m. 6. Calculate the period of the oscillations for each length by dividing the total time by the number of oscillations, 50. Record the values in the appropriate column of your data table. The Simple Pendulum Revised 10/25/2000 5Figure 3. Example of Excel spreadsheet for recording and analyzing data. 7. Graph the period of the pendulum as a function of its length using the chart feature of Excel. The length of the pendulum is the independent variable and should be plotted on the horizontal axis or abscissa (x axis). The period is the dependent variable and should be plotted on the vertical axis or ordinate (y axis). 8. Use the trendline feature to draw a smooth curve that best fits your data. To do this, from the main menu, choose Chart and then Add Trendline . . . from the dropdown menu. This will bring up a Add Trendline dialog window. From the Trend tab, choose Power from the Trend/Regression type selections. Then click on the Optionstab and select Display equations on chart option. 9. Examine the power function equation that is associated with the trendline. Does it suggest the relationship between period and length given by Equation (7)? 10. Examine your graph and notice that the change in the period per unit length, the slope of the curve, decreases as the length increases. This indicates that the period increases with the length at a rate less than a linear rate. The theory and Equation (7) predict that the period depends on the square root of the length. If both sides of Equation 7 are squared then The Simple Pendulum Revised 10/25/2000 622 4 T =gl . (8) If the theory is correct, a graph of T2 versus l should result in a straight line. 11. Square the values of the period measured for each length of the pendulum and record your results in the spreadsheet. 12. Use the chart feature again to graph the period squared, T2, as a function of the length of the pendulum l . The period squared is the dependent variable and should be plotted on the y axis. The length is the independent variable and should be plotted on the x axis. 13. Examine your graph of T2 versus l and check to see if there is a linear relationship between T2 and l so that the data points lie along a line. 14. Use the trendline feature to perform a linear regression to find a straight line that best fits your data points. This time from the Add Trendline dialog window. choose Linear from the Trend/Regression type selections. Click on the Options tab and once again select the Display equations on chart option. This should draw a straight line that best fits the data and should display the equation for this straight line. 15. Equation (8), 22 4 T =gl , is of the form y=ax+b where y=T2, 2 4a = g, x=l , and b=0. A graph of T2 versus l should therefore result in a straight line whose slope, a, is equal to 2 4g. From the equation for the trendline, record the value for the slope, a, and from the equation a=2 4g find g, the acceleration due to gravity. 16. Compare your result with the accepted value of the acceleration due to gravity 9.8 m/s2. Calculate the percent difference in your result and the accepted result. % Difference = [(your result - accepted value)/accepted value] x 100% 17. Using the accepted value of the acceleration due to gravity and Equation 7 calculate the period of a simple pendulum whose length is equal to the longest length measured in Table 1. Compare this theoretical result with the measured experimental result and calculate the percent difference. % Difference = [(Experimental Result - Theoretical Result)/Theoretical Result] x 100%. 18. The equation for the period of a simple period, Equation (7), was developed by assuming that the amplitude is small. The range of amplitudes over which Equation The Simple Pendulum Revised 10/25/2000 7(7) is valid is to be determined by measuring the period of a simple pendulum with different amplitudes. 19. Adjust the length of the pendulum to about 0.6 m. Measure the period of the pendulum when it is displaced 5, 10, 15, 20, 25, 30, 40, 50, and 60 from its equilibrium position. Make a table to record the period T as a function of the amplitude A. 20. Using your data, make a graph of the period versus the amplitude. 21. Measure the length of the pendulum and use Equation (7) to calculate the period of the pendulum. Add this theoretical point to your graph for the period with zero amplitude. 22. Examine your graph for the behavior of the period with amplitude. What conclusions can you draw from your data regarding the range of amplitudes over which Equation (7) is valid? Questions 1. How would the period of a simple pendulum be affected if it were located on the moon instead of the earth? 2. What effect would the temperature have on the time kept by a pendulum clock if the pendulum rod increases in length with an increase in temperature? 3. What kind of graph would result if the period T were graphed as a function of the square root of the length, l . 4. What effect does the mass of the ball have on the period of a simple pendulum? What would be the effect of replacing the steel ball with a wooden ball, a lead ball, and a ping pong ball of the same size?

To measure the resistance of a galvanometer by Kelvins methodTo measure the internal resistance of a Leclanche Cell by a Mances methodTo find the magnetic moment of a magnetTo find the value of the horizontal component of the earths magnetic field at the place of experiment