9 HOOKE S LAW AND SIMPLE HARMONIC MOTION

Transcription

1 Experient 9 HOOKE S LAW AND SIMPLE HARMONIC MOTION Objectives 1. Verify Hoo s law,. Measure the force constant of a spring, and 3. Measure the period of oscillation of a spring-ass syste and copare it with its predicted value Theory How is a spring defored when we applied forces to it? What causes a spring to be different fro another one? How does the period of oscillation of a siple haronic oscillator change, as a function of its ass and the value of its force constant? How does the theoretical value of the period of oscillation of the ass in the spring copare with the easured value? These are soe of the questions that we will be able to answer once we have ade the experient that we will describe shortly. In 1676 Robert Hooe, an English scientist, who lived in Newton s tie, discovered and established the law that taes its nae and that is used to define the elastic properties of a body. In the study of the effects of the forces of tension, and copression, he observed that there was an increase in the length of the spring, or elastic body, which was proportional to the applied force, within certain liits. This observation showed that the deforation is directly proportional to the distorting force, F = x 1 Where F is the force, easureent in newtons (N),, the force constant of the spring and x, the extension, or copression. The negative sign indicates that the force of the string is a restoring force, or opposed to the external force that defors it. This expression is nown as Hooe s law. If the distorting force exceeds a certain axiu value, the body will not return to its original size (or its for) after suppressing that force. Then one says that the body has acquired a peranent distortion. The tension, or copression, saller than the one that produces a peranent distortion is called elastic liit. Hooe s law is not applicable for distorting forces that exceed the elastic liit. On the other hand, when the oveent of an object is repeated in regular intervals, or periods, it is called a periodic otion. If we tae the oscillations fro a siple pendulu, for angular displaceents saller of 1, we have an exaple of periodic otion. Let us consider a particle of ass, attached to a spring that oscillates in the x-direction on a horizontal surface, without friction. See figure 9.1. (Please, go to the following Internet site where there appears the aniation of two siple haronic oscillators with different oscillation frequencies: Figure 9-1 A siple haronic oscillator reacts with a force that opposes the deforation Applying Newton s second law to the spring-ass syste we have: x = a On the other hand, the instantaneous acceleration is defined as, 1

2 d x a = 3 dt Cobining equations and 3 we obtain that: Or, d x = x dt d x dt + x = Equation 5 is a second order linear ordinary differential equation with constant coefficients for which we propose a solution of the for, x (t) = A 0 cos ωt 6 Where A 0 is the aplitude of oscillation, or axiu elongation, and ω, the frequency. This solution is correct if ω = 7 We can see fro here that the period of oscillation, T = ω/π can be written as: Materials T = π 8 Equipent Qty Equipent Qty Force Sensor 1 Syste of spring-ass for Hooe s law 1 Rotational Motion Sensor 1 Base of support 1 Pasco Coputer Interface Metallic rod of 10 centieters in length and 1 diaeter of ½ ". Linear otion accessory 1 Rod with a diaeter of ¾ and 45 c of length 1 ultiple clip 1 Mass and suspension syste 1 Procedure Installation of the equipent 1. Chec that the experient setup on your bench loos lie the one shown in Figure 9- below. Mae sure that the force sensor, the spring, and the linear otion accessory are vertically aligned Part I. Hooe s Law 1. Set the force sensor to zero by pressing the zero button on the sensor while the spring is relaxed. Create a graph of Force (N) versus Position () taing the data force fro the force sensor and the position fro the rotation sensor 3. Press Start. Slowly pull down the rod attached to the lower end of the spring. The diagra of force against spring elongation appears in the graph. Press stop when the lower end of the spring reaches the rotational otion sensor 4. Note: It is very iportant to go slowly when doing the easureent

3 5. Your data ust loo lie the figure to your right: 6. Press the scale button to iprove the graph appearance 7. Press Fitting and select Linear Fitting to obtain the equation for the best straight line that fits your data. The slope of the line is the spring constant. The spring constant for this exaple is 3.71 N/ Figure 9- Experient setup Part II Siple Haronic Oscillator 1. Mae the necessary changes to your setup to arrive to the one shown in the figure to the right. Weigh the spring, and the support, and add a ass of 0,050 ilogra 3. Register the value of the total ass (g) for the first run of this experient 4. Place the otion sensor in the floor, directly underneath the weight support 5. Pull the ass down and release it. Allow it to oscillate a few ties so that the springass syste will ove up and down without uch side to side oveent 6. Mae a graph of Position versus tie by pressing start and stop the easureent after 4 seconds 7. Run the experient again after adding a ass of 0-g (0.00-g) to the support and registering the total ass 8. Repeat the process of data collection Note: The curve of position vs. tie ust reseble a sine function. If this is not the case, verify the alignent between the sensor and the botto of the support 3

4 Data analysis 1. Press the tool Zoo lens to ephasize and extend the first three axiu tips of the curves. Press the button for intelligent tool on the onitor. The intelligent tool will appear in the graph 3. Move the intelligent tool to the first tip in the diagra of position against tie 4. Move the cursor to the inferior corner of the intelligent tool. A triangle (called the tool delta) will appear 5. Drag the delta to the following tip. The value of delta X is the period. In the exaple, the period is Register the period (easured) for the first run 7. Move the intelligent tool to the following tip 8. Use the tool of the delta to find the period between the seconds and third tips 9. Register the period. Calculate an average of these values and register it. 10. Repeat the procedure for the second run of the experient 4

5 Experient 9. Laboratory Report Hooe s Law and siple haronic otion Section Laboratory bench Date: Student s naes: Part I. Hooe s Law 1. Use the table below to write down the slope of the straight line that you obtained fro your graph of Force vs. Position. As your lab instructor the expected value of the spring constant provided by the anufacturer, and write it down in the sae table 3. Calculate the uncertainty % (relative percentile difference) between these two values Value of fro the slope (N/) Expected value of (N/) % Expected value - Measured value % = 100 Expected value Questions 1. Identify the ain sources of error in this part of the experient 5

6 . Write down the atheatical equation that relates the force applied to the spring to its elongation, nown as Hooe s law Proble 1. A person standing up on a bath spring scale reads a weight of 670 N. It is nown that in this case the spring s copression is equal to 0.79 c. (a) Calculate the spring constant value (b) Find the weight of another person who copresses the spring by 0.34 centieters on the sae scale Part II. Siple Haronic Motion 1. Fill out the blans with your data in the following table: Run Total ass, = support + added (g) Average period, T (s) First Second. Mae a graph of T vs.. Notice that you will obtain a straight line with a slope of 4π / since Attach the graph to your report T = π 3. Obtain a value of fro that slope. Copare the value of the spring constant that you just obtained, which is the easured value, to the value provided by your laboratory instructor (theoretical, expected, or reported value). Calculate again the percentile relative uncertainty of these two values Reported value of - Measuredvalue of % = 100 = Reported value of 6

7 Questions 1. Is there a systeatic error between the easured value of and its reported value? Explain your answer referring to the % value that you obtained fro the second part of this laboratory exercise. Suppose you have a siple haronic oscillator with a period T. Find the new value of the period for this siple haronic oscillator if you double its ass 3. Do your results support the hypothesis? Explain your answer Proble A spring stretches by 0,00 when an object of 3.7-g is suspended fro its lower end. How uch ass is required to hang fro this spring so that its frequency of oscillation becoes 4.0 Hz? 7

8 Experient 9. Questions Hooe s law and siple haronic oscillator This questionnaire has soe typical questions on experient 9. All students who are taing the laboratory course of University Physics I ust be able to correctly answer it before trying to ae the experient 1. What is the relation between the force produced by a spring and its elongation? a. The force increases when stretching it b. The force diinishes when stretching it c. The force is the negative of the elongation d. There is no relationship between the e. The force does not change when stretching it. When a force produces a peranent deforation in a spring, a. Hooe s law loses its validity b. a siple haronic oscillator is created c. Hooe s law continues being valid d. the force is proportional to half of the deforation e. the force is proportional to the position 3. In a siple haronic oscillator, when we plot the square of the period versus the ass, which of the following expressions is equal to the value of the spring constant? π a. b. c. d. 4π slope 4π 4π slope e. = slope 4. The atheatical relationship between the period and the frequency of a siple haronic oscillator is that of: a. siple proportionality b. direct proportionality c. inverse proportionality to the square d. direct proportionality to the square e. inverse proportionality 5. A person who weighs 500 N rests on a balance. The spring of the balance is copressed by 0,39 c. What is the value of the spring constant? a N/ b N/ c N/ d N/ e. We need to now the value of the acceleration of the gravity to ae the calculation 6. What is the angular frequency of a siple haronic oscillator whose ass of 5 g oscillates with a spring whose is N/? a. 3,400 rad/s b rad/s c rad/s d Hz e rad/s 8

9 7. The spring constant of a siple haronic oscillator that oscillates with a frequency of 0.5 Hz when its ass is 1.0 g is: a. π N/ b. 0,5 N/ c. π N/c d. 0.5 N/c e. (0.5/9.81) g/ 8. The period of a siple haronic oscillator is defined as: a. The inverse of its angular frequency b. The tie elapsed for a coplete oscillation c. T = d. The tie it taes to reach a coplete extension e. The nuber of oscillations by tie unit 9. The period of a siple haronic oscillator with a of N/ and a ass of 5 g is: a s b. 5 s c s d s e. Unnown 10. The frequency of a siple haronic oscillator with a = N/ and a ass of 5 g is: a. 541 Hz b. a ystery c. 9.8 Hz d Hz e Hz 11. A spring extends when an object of.8 g is suspended fro its end. How uch ass ust support this spring so that its frequency of vibration is 3.0 Hz? a. All that it is able to hold b g c. 3. g d. 7.6 g e. 4.3 g 1. If a siple haronic oscillator has an angular frequency of 9.4 rad/s, what is its frequency in Hz? a. 1.5 Hz b. 3 Hz c. 9.4 Hz d. The sae one, because 1rad/1s = 1 Hz e. 10 Hz 9