EXPERIMENT 14 Simple Harmonic Motion - OpenStudyassets.openstudy.com/...nincompoop-1383298782943-simpleharmoni… · Hooke’s law, using rubber-band and spring elongations. ... EXPERIMENT

221

OVERVIEW

Experiment 14 considers simple harmonic motion (SHM) with TI and/or CI procedures. The TI procedure examines Hooke’s law, using rubber-band and spring elongations. Simple harmonic motion is investigated through the period of oscillation of a mass on a spring.

The CI procedure investigates the SHM of a simple pendulum and the resulting conversion of energy (kinetic and potential) that occurs during the motion. An electronic sensor measures the angular speed, v 5 Du/Dt, of the pendulum, from which the tangential speed is computed and the energies calculated.

E X P E R I M E N T 1 4

Simple Harmonic Motion

INTRODUCTION AND OBJECTIVES

Elasticity implies a restoring force that can give rise to vibrations or oscillations. For many elastic materials, the restoring force is proportional to the amount of deformation, if the deformation is not too great.

This is best demonstrated for a coil spring. The restoring force F exerted by a stretched (or compressed) spring is proportional to the stretching (compressing) distance x, or F ~ x. In equation form, this is known as Hooke’s law,

F 5 2kx

where x is the distance of one end of the spring from its unstretched (x 5 0) position, k is a positive constant of proportionality, and the minus sign indicates that the dis-placement and force are in opposite directions. The con-stant k is called the spring (or force) constant and is a relative indication of the “stiffness” of the spring.

A particle or object in motion under the infl uence of a linear restoring force, such as that described by Hooke’s law, undergoes what is known as simple harmonic motion (SHM). This periodic oscillatory motion is one of the common types found in nature.

In this experiment, Hooke’s law will be investigated, along with the parameters and description of simple har-monic motion.

After performing this experiment and analyzing the data, you should be able to:

OBJECTIVES

1. Tell how Hooke’s law is represented graphically and cite an example of an elastic object that does not follow Hooke’s law.

2. Explain why simple harmonic motion (SHM) is simple and harmonic.

3. Better understand how the period of a mass oscillat-ing on a spring varies with the mass and the spring constant.

OBJECTIVES

1. Explain the energy conversion that happens during the simple harmonic motion of a pendulum.

2. Experimentally verify the law of conservation of mechanical energy.

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EQUIPMENT NEEDED

Coil spring• Wide rubber band• Slotted weights and weight hanger•

Laboratory timer or stopwatch• Meter stick• Laboratory balance• 2 sheets of Cartesian graph paper•

T I E X P E R I M E N T 1 4

Simple Harmonic Motion

THEORY

A. Hooke’s Law

The fact that for many elastic substances the restor-ing force that resists the deformation is directly pro-portional to the deformation was fi rst demonstrated by Robert Hooke (1635–1703), an English physicist and contemporary of Isaac Newton. For one dimension, this relationship, known as Hooke’s law, is expressed math-ematically as

F 5 2kDx 5 2k(x 2 xo) (TI 14.1)

or

F 5 2kx (with xo 5 0)

where Dx is the linear deformation or displacement of the spring and xo is its initial position. The minus sign indicates that the force and displacement are in opposite directions.

For coil springs, the constant k, is called the spring or force constant. The spring constant is sometimes called the “stiffness constant,” because it gives an indication of the relative stiffness of a spring—the greater the k, the greater the stiffness. As can be seen from TI Eq. 14.1, k has units of N/m or lb/in.

According to Hooke’s law, the elongation of a spring is directly proportional to the magnitude of the stretching force.* For example, as illustrated in ● TI Fig. 14.1, if a spring has an initial length yo, and a suspended weight of mass m stretches the spring to a length y1, then in equilib-rium the weight force is balanced by the spring force and

F1 5 mg 5 k(y1 2 yo)

Here y is used to indicate the vertical direction, instead of x as in TI Eq. 14.1, which is usually used to mean the hori-zontal direction. Similarly, if another mass m is added and the spring is stretched to a length y2, then

F2 5 2mg 5 k 1y2 2 yo 2and so on for more added weights. The linear relation-ship of Hooke’s law holds, provided that the deformation or elongation is not too great. Beyond the elastic limit, a spring is permanently deformed and eventually breaks with increasing force.

Notice that Hooke’s law has the form of an equation for a straight line:

F 5 k(y 2 yo)

or

F 5 ky 2 kyo

which is of the general form y 5 x 1 b

TI Figure 14.1 Hooke’s law. An illustration in graphical form of spring elongation versus force. The greater the force, the greater the elongation, F 5 2ky. This Hooke’s law relationship holds up to the elastic limit.

*The restoring spring force and the stretching force are equal in magnitude and opposite in direction (Newton’s third law).

In actual practice, the amplitude decreases slowly as energy is lost to friction, and the oscillatory motion is slowly “damped.” In some applications, the simple har-monic motion of an object is intentionally damped, for example, the spring-loaded needle indicator of an electrical measurement instrument or the dial on a common bathroom scale. Otherwise, the needle or dial would oscillate about the equilibrium position for some time, making it diffi cult to obtain a quick and accurate reading.

The period of oscillation depends on the parameters of the system and, for a mass on a spring, is given by

T 5 2pÅmk

(TI 14.3)

(period of mass oscillating on a spring)

EXPERIMENTAL PROCEDURE

A. Rubber-Band Elongation

1. Hang a rubber band on a support and suspend a weight hanger from the rubber band. Add an appropriate weight to the weight hanger (for example, 100–300 g), and record the total suspended weight (m1g) in TI Data Table 1. Fix a meter stick vertically alongside the weight hanger and note the position of the bottom of the weight hanger on the meter stick. Record this as y1 in the data table.

224 EXPERIMENT 14 / Simple Harmonic Motion

B. Simple Harmonic Motion

When the motion of an object is repeated in regular time intervals or periods, it is called periodic motion. Examples include the oscillations of a pendulum with a path back and forth along a circular arc and a mass oscillating lin-early up and down on a spring. The latter is under the infl u-ence of the type of force described by Hooke’s law, and its motion is called simple harmonic motion (SHM)—simple because the restoring force has the simplest form and harmonic because the motion can be described by harmonic functions (sines and cosines).

As illustrated in ● TI Fig. 14.2, a mass oscillating on a spring would trace out a wavy, time-varying curve on a mov-ing roll of paper. The equation for this curve, which describes the oscillatory motion of the mass, can be written as

y 5 A cos

2ptT

(TI 14.2)

where T is the period of oscillation and A is the amplitude or maximum displacement of the mass.

The amplitude A depends on the initial conditions of the system (that is, how far the mass was initially dis-placed from its equilibrium position). If the mass were initially (t 5 0) pulled below its equilibrium position (to y 5 2A) and released, the equation of motion would be y 5 2A cos 2pt/T, which satisfi es the initial condition at t 5 0 with cos 0 5 1 and y 5 2A. The argument of the cosine, (2pt/T), is in radians rather than degrees.

TI Figure 14.2 Simple harmonic motion. A marker on a mass oscillating on a spring traces out a curve, as illustrated, on the moving paper. The curve may be represented as a function of displacement (magnitude y) versus time, such as y 5 A cos 2pt/T, where y 5 A at t 5 0.

EXPERIMENT 14 / Simple Harmonic Motion 225

2. Add appropriate weights (for example, 100 g) to the weight hanger one at a time, and record the total sus-pended weight and the position of the bottom of the weight hanger on the meter stick after each elongation (y2, y3, etc.). The weights should be small enough so that seven or eight weights can be added without over-stretching the rubber band.

3. Plot the total suspended weight force versus elonga-tion position (mg versus y), and draw a smooth curve that best fi ts the data points.

B. Spring Elongation

4. Repeat Procedures 1 and 2 for a coil spring and record the results in TI Data Table 2. Choose appropriate mass increments for the spring stiffness. (A commer-cially available Hooke’s law apparatus is shown in ● TI Fig. 14.3.)

5. Plot mg versus y on the same sheet of graph paper used in Procedure 3 (double-label axes if necessary), and draw a straight line that best fi ts the data. Determine the slope of the line (the spring constant k), and record it in the data table. Answer TI Questions 1 through 3 following the data tables.

C. Period of Oscillation

6. (a) On the weight hanger suspended from the spring, place a mass just great enough to prevent the spring from oscillating too fast and to prevent the hanger from moving relative to the end of the spring during oscillations when it is pulled down (for example, 5 to 10 cm) and released. Record the total mass in TI Data Table 3.

(b) Using a laboratory timer or stopwatch, release the spring weight hanger from the predetermined ini-tial displacement and determine the time it takes for the mass to make a number (5 to 10) of com-plete oscillations or cycles.

The number of cycles timed will depend on how quickly the system loses energy or is damped. Make an effort to time enough cycles to get a good average period of oscillation. Record in the data table the total time and the number of oscillations.

Divide the total time by the number of oscillations to determine the average period.

7. Repeat Procedure 6 for four more mass values, each of which is several times larger than the smallest mass, and record the results in TI Data Table 3. The initial displacement may be varied if necessary. (This should have no effect on the period. Why?)

8. Plot a graph of the average period squared (T 2) ver-sus the mass (m), and draw a straight line that best fi ts the data points. Determine the slope of the line and compute the spring constant k. [Note from TI Eq. 14.3that k is not simply equal to the slope; rather, k 5 (2p)2/slope.]

Compare this value of k with that determined from the slope of the spring elongation graph in Part Bby computing the percent difference, and finish answering the TI Questions.

TI Figure 14.3 Hooke’s law apparatus. The variables of Hooke’s law (F 5 mg and x) are measured using spring elongation. (Photo Courtesy of Sargent-Welch.)

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Name Section Date

Lab Partner(s)

T I E X P E R I M E N T 1 4

Simple Harmonic Motion Laboratory Report

A. Rubber-Band Elongation

DATA TABLE 1

Total suspended weight* ( )

Scale reading ( )

m1g y1

m2g y2

m3g y3

m4g y4

m5g y5

m6g y6

m7g y7

m8g y8

* It is convenient to leave g, the acceleration due to gravity, in symbolic form; that is, if m15100 g or 0.100 kg, then weight 5 m1g 5 (0.100 kg)g N,but your instructor may prefer otherwise. Be careful not to confuse the symbol for acceleration due to gravity, g (italic), with the abbreviation for gram g (roman).

B. Spring Elongation

DATA TABLE 2

Total suspended weight* ( )

Scale reading ( )

m1g y1

m2g y2

m3g y3

m4g y4

m5g y5

m6g y6

m7g y7

m8g y8

* It is convenient to leave g in symbol form, even when graphing.

Calculations(show work)

k (slope of graph) ___________________ (units)

Don’t forget units(continued)

CNSP

20.

228

E X P E R I M E N T 1 4 Simple Harmonic Motion Laboratory Report

C. Period of Oscillation

DATA TABLE 3

Total suspended mass ( )

Total time( )

Number ofoscillations ( )

Average periodT

T 2

( )

m1

m2

m3

m4

m5

Calculations(show work) Slope of graph ___________________

Computed spring constant k ___________________

Percent difference (of k’s in B and C) ___________________

QUESTIONS

1. Interpret the intercepts of the straight line for the spring elongation in the mg-versus-y graph of Part B.

Name Section Date

Lab Partner(s)

229


2. Is the elastic property of the rubber band a good example of Hooke’s law? Explain.

3. Draw a horizontal line through the y-intercept of the straight-line graph of Part B, and form a triangle by drawing a vertical line through the last data point.(a) Prove that the area of the triangle is the work done in stretching the spring. (Hint:

W 5 12 kx2, and area of triangle A 5 1

2 ab, that is, 12 the altitude (a) times the base (b).)

(b) From the graph, compute the work done in stretching the spring.

4. Interpret the x-intercept of the straight line of the T 2-versus-m graph of Part C.

(continued)

230


5. For a mass oscillating on a spring, at what positions do the (a) velocity and (b) acceleration of the mass have maximum values?

6. What is the form of the equation of motion for the SHM of a mass suspended on a spring when the mass is initially (a) released 10 cm above the equilibrium position; (b) given an upward push from the equilibrium position, so that it undergoes a maximum displacement of 8 cm; (c) given a downward push from the equilibrium position, so that it undergoes a maximum displacement of 12 cm? (Hint: Sketch the curve for the motion as in TI Fig. 14.2 and fi t the appropriate trigonometric function to the curve.)

7. For case (a) in Question 6 only, what is the displacement y of the mass at times (a) t 5 T/2; (b) t 5 3T/2; (c) t 5 3T?

231

but is proportional to the sin u instead. A pendulum can be approximated to be in SHM motion only if the angle u is small, in which case sin u < u (where u is in radians). Thus,

F 5 mg sin u < mg u (CI 14.2)

Notice that in this approximation, the force is directly pro-portional to the displacement u.

As the pendulum swings, kinetic energy is converted into potential energy as the pendulum rises. This potential energy is converted back to kinetic energy as the pendu-lum swings downward. The kinetic and potential energies of the pendulum at any moment during its motion can eas-ily be determined. The kinetic energy of a pendulum of mass m moving with a linear speed v is given by

K 5 12 mv2 (CI 14.3)

The potential energy, measured with respect to the equilib-rium position, depends on the height above the equilibrium at a particular time. That is,

U 5 mgh 5 mg(L 2 L cos u) (CI 14.4)

(See ● CI Fig. 14.2.)

THEORY

In this experiment, the simple harmonic motion of a pen-dulum will be investigated by examining the energy con-versions that occur during the motion. Simple harmonic motion is the motion executed by an object of mass m subject to two conditions:

The object is subject to a force that is proportional • to the displacement of the object that attempts to restore the object to its equilibrium position.No dissipative forces act during the motion, so there • is no energy loss.

Notice that as it is described in theory, simple harmonic motion is an idealization because of the assumption of no frictional forces acting on the particle.

In this experiment, the simple harmonic motion of a pendulum will be investigated. A simple pendulum consists of a mass (called a bob) suspended by a “massless” string from a point of support. The pendulum swings in a plane.

The restoring force on a simple pendulum is the com-ponent of its weight that tends to move the pendulum back to its equilibrium position. As can be seen from ● CI Fig. 14.1, the magnitude of the force is

F 5 mg sin u (CI 14.1)

Note, however, that this force is not proportional to the angu-lar displacement u of the pendulum, as required for SHM,

C I E X P E R I M E N T 1 4

Simple Harmonic Motion EQUIPMENT NEEDED

Rotary Motion Sensor (PASCO CI-6538)• Mini-rotational accessory (PASCO CI-6691. This • set includes a brass mass and a light rod to make the pendulum.)

!

mg

!

T

mg sin !

mg cos !

CI Figure 14.1 Forces acting on a swinging pendulum. The restoring force acting on a pendulum is the component mg sin u of gravity, which attempts to bring the pendulum back to the equilibrium position.

L cos !

L ! L cos !

!

L

CI Figure 14.2 The elevation of a pendulum with respect to the equilibrium position. The elevation of a pendu-lum with respect to the equilibrium (lowest) position can be expressed in terms of L, the length of the pendulum, and of u, as L 2 L cos u. (The angular displacement has been exaggerated in the illustration. For simple harmonic motion, u must be small.)

Support rods and clamps•


In this experiment, a sensor will keep track of the angular position, u, of the pendulum as it swings. The sen-sor will also keep track of the angular speed, v 5 Du/Dt, of the pendulum. The linear speed (v) can then be deter-

mined as v 5 vL, where L is the length of the pendulum and also the radius of the circular arc described by its motion. The kinetic and potential energies of the pendu-lum at any time can then be calculated.

BEFORE YOU BEGIN

1. Measure the mass of the pendulum bob (M) and record it in the laboratory report, in kilograms.

2. Measure the length of the pendulum (L), in meters, from the center of rotation to the center of the bob. Record it in the report.

This information will be needed during the setup of Data Studio.

SETTING UP DATA STUDIO

1. Open Data Studio and choose “Create Experiment.” 2. The Experiment Setup window will open and you will

see a picture of the Science Workshop interface. There are seven channels to choose from. (Digital Channels 1, 2, 3, and 4 are the small buttons on the left; ana-log Channels A, B, and C are the larger buttons on the right, as shown in ● CI Fig. 14.3.)

3. Click on the Channel 1 button in the picture. A win-dow with a list of sensors will open.

4. Choose the Rotary Motion Sensor from the list and press OK.

5. The diagram now shows you the properties of the RMS sensor directly under the picture of the interface. (See CI Fig. 14.3.)

6. Connect the sensor to Channels 1 and 2 of the inter-face, as shown on the computer screen.

7. Adjust the properties of the RMS as follows:First Measurements tab: select Angular Position, Chapters 1 and 2, and select the unit of measure to be degrees. Also select Angular Velocity, Channels 1 and 2, in rad/s.Rotary Motion Sensor tab: set the Resolution to high (1440 divisions/rotations); and set the Linear Scale to Large Pulley (Groove).

CI Figure 14.3 Experimental setup. The seven available channels are numbered 1 through 4 and A, B, or C. The rotary motion sensor, connected to Channels 1 and 2, will measure the angular position and the angular velocity of the pendulum. Make sure that the angular position is being measured in degrees, but the angular velocity in rad/s. (Reprinted courtesy of PASCO Scientifi c.)


10. Calculation of the linear speed:(a) In the same calculator window, clear the defi ni-

tion box and enter the following equation:

V 5 L * smooth (6, w)

This is the calculation of the linear speed v 5 vL, which will be called V. Note that the length L of the pendulum is multiplied by the angular speed, which is called w here. The smooth function is to produce a sharper graph.

(b) Press the Accept button after entering the for-mula. The variables L and w will appear in a list. L will have the value defi ned before, but w will be waiting to be defi ned.

(c) To define the variable w, click on the drop menu button on the left side of the variable. A list of op-tions will show, asking what type of variable this is.

• Define w as a Data Measurement and, when prompted, choose Angular Velocity (rad/s).

11. Calculation of the kinetic energy:(a) Still in the same calculator window, press the

New button again to enter a new equation.(b) Clear the defi nition box and enter the following

equation: KE 5 0.5 * M * v2. This is the calcula-tion of the kinetic energy K 5 1

2 mv2, that will be called KE.

(c) Press the Accept button after entering the for-mula. The variables M and v will appear in a list;

Set the Sample Rate to 20 Hz.The Data list on the left of the screen should now have two icons: one for the angular position data, the other for the angular velocity data.

8. Open the program’s calculator by clicking on the Calculate button, on the top main menu. Usually, a small version of the calculator opens, as shown in ● CI Fig. 14.4. Expand the calculator win-dow by clicking on the button marked Experiment Constants.

9. The expanded window (shown in ● CI Fig. 14.5) is used to establish values of parameters that will remain constant throughout the experiment. In this case, these are the length of the pendulum (L) and the mass of the pendulum (M), which have already been measured. This is how to do it:(a) Click on the lower New button (within the

“Experiment Constants” section of the calculator window) and enter the name of the constant as L, the value as the length of the pendulum measured before, and the units as meters (m).

(b) Click the lower Accept button.(c) Click on the New button again, and enter the

name of the constant as M, the value as the mass of the pendulum measured before, and the units as kilograms (kg).

(d) Click the lower Accept button.(e) Close the experiment constants portion of the

calculator window by pressing the button marked Experiment Constants again.

CI Figure 14.4 The calculator window. This small version of the calculator window opens when the Calculate button is pressed. The calculator will be used to enter equations that handle the values measured by the sensor. The computer will perform the calculations automatically as the sensor takes data. (Reprinted courtesy of PASCO Scientifi c.)


(b) Clear the defi nition box and enter the follow-ing equat ion: PE 5 M * 9.81 * (L 2 L * cos (smooth (6, x))). This is the calculation of the potential energy U 5 mgh 5 mg(L 2 L cos u), which will be called PE. Note that M is the mass, 9.81 is the value of g, and the variable x in this formula will stand for the angular posi-tion u of the pendulum, in degrees.

(c) Press the button marked DEG that is under the definition box. This will make sure the calcu-lation of the cosine is done in degrees, not in radians.

M is the value entered before for the mass, and v is waiting to be defi ned.

(d) To defi ne the variable v, click on the drop menu button on the left side of the variable. The list of options will show, asking what type of variable this is.

• Define v as a Data Measurement and, when prompted, choose V, the equation defined previously.

12. Calculation of the potential energy:(a) Press the New button once again to enter a new

equation.

(a)

(2) Enterconstantsymbol.

L New Remove Accept

Value 0.35 mUnits

(1) Press New.

(4) Accept.

(b)

(3) Enter the valueof this constantand the units.

Experiment Constants!

CI Figure 14.5 The expanded calculator window. (a) After the button marked Experiment Constants is pressed, the calculator window expands to full size. (b) The “Experiment Constants” section is the lower part of the expanded calculator window. This section is used to defi ne parameters that are to remain constant during the experiment. The diagram shows the steps needed to enter experimental constants into the calculator. (Reprinted courtesy of PASCO Scientifi c.)


17. Repeat step 15 to create a second graph window. Graph 2 will also be a graph of angular position (deg) versus time.

18. Drag the PE equation icon and drop it on Graph 2. Graph 2 will then split in two, showing both the posi-tion and the PE of the pendulum at any time t, with matching time axes.

19. It is not necessary to be able to see both graph windows at the same time, but they can be moved around the screen so that both are visible. Their sizes may also be adjusted so that when they are active, they occupy the full screen individually. It is easy to change from view-ing one to viewing the other by clicking on the particular graph to bring it to the front. ● CI Fig. 14.6 shows what the screen will look like after all the setup is fi nished.

EXPERIMENTAL PROCEDURE

1. Put the rotary motion sensor on a support rod. Install the mass on the light rod, and then install the pendulum on the front screw of the rotary motion sensor. A diagram of the equipment setup is shown in ● CI Fig. 14.7.

(d) Press the Accept button after entering the equa-tion. The variables M, L, and x will appear in a list, with x waiting to be defi ned.

(e) Define x as a Data Measurement and, when prompted, choose Angular Position (deg).

(f) Press the Accept button.

13. Close the calculator window.

14. The data list on the upper left of the screen should now include icons for the three quantities that are cal-culated: V, KE, and PE. A small calculator icon will show on the left of the calculated data quantities.

15. Create a graph by dragging the Angular Position (deg) data icon and dropping it on top of the “Graph” icon on the displays list. A graph of angular position (deg) versus time will open. The window will be called Graph 1.

16. Drag the KE equation icon and drop it somewhere on top of the graph created in step 15. The graph will then split in two, with the graph of angular po-sition versus time on top and the graph of KE versus time on the bottom. The graphs will have matching time axes.

CI Figure 14.6 Data Studio setup. Graph displays are generated for angular position, kinetic energy, and potential energy. The individual graph windows can be viewed together (as in this picture) or independently, if resized to fi t the full screen. (Reprinted courtesy of PASCO Scientifi c.)


3. After pressing the start button, displace the pendulum a small angle (#10°) to the side and let it go.

4. Collect data for about 5 or 6 seconds, and then press the STOP button.

5. Print the graphs and paste them to the laboratory report.

6. Read from any of the position graphs what was the maximum amplitude of the pendulum, and record it in CI Data Table 1.

7. Determine from the graph the period of oscillation of the pendulum, and record it in the table.

8. From the kinetic energy graph, look at the fi rst clear complete cycle of the motion, and fi nd the maximum kinetic energy during that cycle. Record it in the table. Record also the position of the pendulum when the maximum kinetic energy was reached.

9. From the potential energy graph, look at the fi rst clear complete cycle of the motion, and fi nd the maximum potential energy during that cycle. Record it in the table. Record also the position of the pendulum when the maximum potential energy was reached.

10. Repeat for the minimum values of kinetic and poten-tial energies.

11. To further reinforce the idea of conversions between kinetic and potential energy, create a new graph (“Graph 3”) by dragging the kinetic energy data icon and dropping it on top of the “Graph” icon on the displays list. Then drag the potential energy icon and drop it in the graph. This graph will show both KE and PE as functions of time.

2. The rotary motion sensor will set its “zero” at the location of the pendulum when the START button is pressed. If we want the position u 5 0 to correspond with the equilibrium position of the pendulum, it is very important that the START button be pressed while the pendulum is at rest in the equilibrium position.

Centerscrew

Lightrod

Pendulumbob

Supportrod

Rotary motionsensor

CI Figure 14.7 The experimental setup. The light rod with the bob at the end is attached to the front screw of the rotary motion sensor.

237

Name Section Date

Lab Partner(s)

C I E X P E R I M E N T 1 4

Simple Harmonic Motion Laboratory Report

DATA TABLE 1

Purpose: To examine the variations of kinetic and potential energy as a pendulum swings.

Mass of pendulum, M kg Max. amplitude 8

Length, L m Period s

ValuePosition of pendulum

(deg)

KE max

PE max

KE min

PE min

Don’t forget to attach the graphs to the laboratory report.

Don’t forget units(continued)

238


QUESTIONS

1. Compare the values of the maximum kinetic energy and the maximum potential energy. Discuss them in terms of the conservation of energy.

2. The following diagram illustrates three different positions of the pendulum as it moves in simple harmonic motion. (The angular displacement has been exaggerated for illustration purposes.) Label in the diagram which position corresponds to maximum KE, which to maximum PE, which to minimum KE, and which to minimum PE.

3. Was the amplitude of the pendulum constant? Explain.

4. The period of a simple pendulum in SHM is given by T 5 2p#Lg. Use the measured

length of the pendulum to calculate its period using this formula. Then compare to the period you determined from the graph. Discuss what causes the percent error.

5. Optional Exercise: Create a new calculation (in the calculator window) that will determine the total energy of the pendulum. That is, calculate KE 1 PE. Then plot the total energy as a function of time. Was the total energy constant? Explain.

239

Name Section Date

Lab Partner(s)

E X P E R I M E N T 1 5

Standing Waves in a String Advance Study Assignment

Read the experiment and answer the following questions.

1. How is wave speed related to frequency and wavelength? How is the period of oscillation related to wave speed?

2. What is a standing wave, and what are nodes and antinodes?

3. What are normal modes?

(continued)

240

E X P E R I M E N T 1 5 Advance Study Assignment

4. How does the wavelength of a standing wave in a vibrating string vary with the tension force in the string and/or the linear mass density of the string?

5. Standing waves in a string can be produced by oscillating the string at the various natural frequencies. However, in this experiment the string vibrator has only one frequency. How, then, are standing waves with different wavelengths produced?

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EXPERIMENT 14 Simple Harmonic Motion - OpenStudyassets.openstudy.com/...nincompoop-1383298782943-simpleharmoni… · Hooke’s law, using rubber-band and spring elongations. ... EXPERIMENT