Simple Harmonic Motion

Physics 211 Experiment #12 Simple Harmonic Motion Experiment In this experiment, a motion sensor is used to measure the position of an oscillating mass as a function of time. The frequency of oscillations will be obtained by measuring the velocity and acceleration of the oscillations, and fitting the data to a sine function. The dependence of oscillation period on the mass applied and on the spring constant will be studied. Introduction An object oscillating in simple harmonic motion is described by

y  A sin  2 ft     A sin   t   

(1)

where: y = distance from the equilibrium position at time t A = amplitude = maximum distance from equilibrium position f = frequency = number of oscillations per second. An oscillation is one complete back-and-forth motion  = angular frequency of the oscillation = 2f  = initial phase angle T = The period of the oscillation, T 

1 . f

dy = the velocity of the mass =  A cos   t    . dt d2y 2 = the acceleration of the mass =  A sin   t    . 2 dt Theory When a mass hangs from a (massless) spring and oscillates vertically, its period is

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T  2

m k

where

(2)

m = mass hanging from spring k = spring constant (k = force/elongation) Squaring both sides,

T 2  4 2

m k

If the spring’s mass is not negligible this becomes (for a uniform spring)

T 2  4 2

m  13 mspring k

,

which can be written

 m T  4  4 2   k  2

2

1 3

mspring  . k 

(3)

For a particular spring, this relation of period squared to mass can be written as a linear equation

y  Cx  D where y  T and x = mass. 2

So a graph of T2 versus mass should be a straight line with Slope = C 

4 2 k

Intercept = D 

(4)

4 2 k



1 3

mspring



(5)

Pre lab assignment 1. Find the period and the frequency of an object that oscillates 30 times in 44 seconds.

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2. In the sample graph, find the value of each of the following quantities; make sure you include proper units! amplitude Simple Harmonic Motion 0.1 frequency maximum velocity maximum acceleration 0.05 initial phase angle (of positiontime graph) Displacement (meters)

y( t)

3. In the sample graph, at t  0.9 seconds, y = maximum. Explain why v=0 a = negative maximum

0

0.05

0.1

0

0.5

Also, at t  1.2 sec., y is at the midpoint of its oscillation. Explain why v = negative maximum a=0

1

1.5

2

2.5

3

t Time (seconds)

Apparatus Pasco 750 Interface Motion sensor Spring, 6 cm by 1.5 cm from Pasco track accessories Large table clamp, right angle clamp, multi-position pendulum clamp and rods to hold spring and motion sensor (see Figure 1) 50 gram mass holder 50 grams of masses (1x10 gram and 2x20 gram masses) Meterstick

Procedure and Analysis for the Simple Harmonic Motion Experiment I. Set-up of computer and interface 1. Turn on the Pasco 750 interface first. Verify that the indicator light is on. 2. Turn on the computer and login. 3. Set up the equipment, as shown in Figure 1 4. Click on Data Studio, following separate Data Studio instructions.

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5. Select Motion Sensor. 6. Double click on Motion to get to Sensor Properties. 7. Under Motion Sensor, increase trigger rate to 25 Hz. 8. Under Measurement, deselect position, and leave velocity and acceleration selected. 9. Click and drag velocity from the Data Window, to the graph icon to create a velocity versus time graph. 10. Click and drag acceleration from the Data Window to the bottom of the velocity graph to create an acceleration graph below the velocity graph. 11. Click on the lock icon to keep the time axes of the plots locked together.

II.

Set-up of equipment Set-up a desk clamp and rods to hold the spring as in Figure 1. Hang the 50 gram weight holder from the spring, as shown in Figure 1.

III.

Measurement of elongation of spring versus applied force Hold the meterstick in a vertical position next to the weight holder, with the 100 centimeter end touching the floor. Read the position of the bottom of the weight holder; record the total mass on the spring and the position of the mass into an excel spreadsheet. Add 10 grams to the holder and again read and record the mass and position into the excel spreadsheet. Repeat until the total hanging mass is 100 grams (the 50 gram holder plus 50 grams of weights.) This data will be graphed later to determine the spring constant of the spring.

IV. Set-up of the motion sensor

Figure 1. Spring, hanging mass, motion sensor, and miscellaneous rods and clamps for the SHM experiment

Use the desk clamp and rods to set up the motion sensor as in Figure 1. The motion sensor should be resting on the floor directly below the weight holder, and should be leveled so that its beam goes directly upward. Set the beam width switch on the sensor for narrow beam and connect the yellow plug to digital channel 1 of the Pasco interface, and the other plug to channel 2.

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V. Recording of position-time data during oscillations With just the 50 gram holder on the spring, raise or lower the rod holding the spring until the bottom of the weight holder is about 40 centimeters above the motion sensor. This is done so that the distance from sensor to weight holder will never be less than about 30 centimeters during an oscillation. This is to insure that the motion sensor accurately measures the distance. Start the weight holder oscillating vertically, about 5 centimeters above and below the equilibrium position. Click on START to begin recording. After a minimum of 5 oscillations, click on STOP. VI.

Determining the oscillation frequency by a sinusoidal fit 1. Click on Zoom Select to select the data to be fit. Go to Fit, and select Sine Series Fit. Fit the velocity data, and the acceleration data separately. The data points should form a smooth sine curve. If they don’t, delete the data and record data again. To delete the data, click on run#1 in the experiment set-up window, hit delete, and click on OK. 2. The fitted curve should match the data; if it does not fit well, ask the instructor to assist you. 3. Into a second excel spreadsheet, record the mass on the spring, the amplitude of the velocity, the amplitude of the acceleration, the frequency of the velocity, and the frequency of the acceleration. 4. Print out a few representative graphs to be included with your laboratory report. 5. Increase the hanging mass to 60 grams (total) and again adjust the spring support so that the mass hanger is about 50 cm above the motion sensor. Repeat V and VI. 6. Repeat the above steps for a total mass of 70, 80, 90, and 100 grams. (Print a graph for each successful run.) 7. Finally, disassemble the apparatus and measure the mass of the spring on a balance. Record this number.

VII.

Calculations 1. Determine the spring constant by plotting the data from part III. Plot a graph of Force versus x and use linear regression to find the slope of the best fit straight line. This is the spring constant k, since

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F  kx F  k x F k  slope x 2. Determine the spring constant k using data from the oscillating masses. First calculate the period of oscillation: Use the relation, T 

1 and obtain f

the frequency from the fit to the velocity data. Calculate T2, and plot T2 versus mass m. Fit the data with a linear trend line, and determine the slope and y-intercept of the fit. Use equation (4) to obtain a second determination of the spring constant k. Compare these two values of K. 3. Compare the determinations of k from step 1 and step 2 of the calculations, and present the percent difference between the two. Also, compare the predicted (equation (5)) and experimentally determined values of the yintercept, and discuss the percent difference between them. 4. Predict the value of the intercept of the graph obtained in step 2 by calculating D using equation 5, the measured mass of the spring, and the value of k obtained from the graph in step 2. Is the value of D equal to the intercept you determined in step 2. What is the percent difference between them? Discuss the result in your report. 5. For every oscillating hanging mass, is the amplitude of the acceleration graph divided by the amplitude from the velocity graph equal to the angular frequency of oscillation?. 6. For each mass, compare the frequency determined from the velocity and acceleration data, and compute the percent difference

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