Physics 211: Lab Oscillations. Simple Harmonic Motion.

Transcription

1 Physics 11: Lab Oscillations. Siple Haronic Motion. Reading Assignent: Chapter 15 Introduction: As we learned in class, physical systes will undergo an oscillatory otion, when displaced fro a stable equilibriu. Moreover, if the initial displaceent is sall, the oscillations will be haronic, i.e. a physical quantity x( describing the state of the syste will be governed by the following differential equation: d x( + ω x( = 0, (1) dt where the angular frequency ω is a constant deterined by the paraeters of the syste but does not depend on the initial displaceent. The ost generic solution to this equation is given by a cos-function x = Acos( ω t + θ ), () ( 0 where the aplitude A and the initial phase θ are deterined by the initial conditions. Note that the quantity x is not necessarily displaceent, it ay be any physical quantity associated with the syste, for instance, the electric current, teperature, concentration of a cheical substance or even a stock price on the arket. In this lab we shall restrict ourselves to two particular echanical oscillatory systes: the atheatical pendulu and a spring oscillator depicted in the figures below. l φ k Matheatical pendulu Spring oscillator The frequency of the spring oscillator was derived in class and is given by: Frequency of the spring oscillator k ω =. (3) 008 Penn State University Physics 11: Lab Oscillations. Siple Haronic Motion.

2 In Activity II you will experientally check this forula. We now turn to deriving the frequency (and hence the period) of the atheatical pendulu. There are, of course, ore than one way of deriving it. Let us, for a change, consider the angular ethod. That is we characterize the oscillations by the deflection angle φ. The oscillations occur around the stable equilibriu which corresponds to φ=0, i.e. the botto point. Viewing the oscillations as rotational otion around the pivot point, where the string is attached to ceiling, we can write down Newton s second law in its angular for: τ = Iα. (4) There are two forces acting on the bob: tension and force of gravity, but the forer one has a zero ar length. Therefore the net torque is produced by gravity and equals (CHECK!) τ = gl sinϕ. (5) (What does the inus sign ean here?). The rotational inertia of the bob, which can be treated as a point particle, is siply I=l. Writing the angular acceleration as the second tie derivative of the angle and using Eq.(5), Newton s second law (4) becoes: d ϕ τ gl sinϕ g sinϕ α = =. (6) dt I l l Coparing the very beginning of the equation with its very end, and bringing both ters on the lefthand side of the equation results in d ϕ g + sinϕ = 0. (7) dt l This equation is the exact equation of otion which is valid for any deflection angle. The solution is coplicated and associated with elliptic functions. However in the case sall oscillations: ϕ << π /, the sin-function can be very well approxiated by the angle itself, which yields the following equation d ϕ( g + ϕ( = 0, (8) dt l which clearly has the for of the siple haronic otion, Eq. (1). Here the oscillating quantity is the angle ϕ, and the coefficient in front of ϕ ( should be identified with ω. Therefore the angular frequency of the atheatical pendulu is given by 008 Penn State University Physics 11: Lab Oscillations. Siple Haronic Motion.

3 g ω =, (9) l as we had hand-wavingly obtained in class based on units analysis. Note that the frequency does not depend on the ass of the bob. In Activity I you shall use this forula to easure the acceleration of gravity. Equipent List: Stand with vertical rod Long string and weight Stopwatch Low-friction track with clap Two springs with different k Cart and additional bar asses Motion Sensor Siple Haronic Motion. Lab Activity I: Matheatical Pendulu Goals: Deterine the period and angular frequency of a atheatical pendulu. Check how the period depends on the aplitude of the oscillations. Using the relation between the frequency and the acceleration of gravity find g with sufficient precision. Suspend the weight fro the string provided. Note that a longer string results in a greater period of oscillations, hence allows for a better accuracy when easuring tie. Deflect the bob fro the vertical by a sall angle (no greater than ) and let it oscillate. 1. Explain how to get a better accuracy when easuring the period.. Record the value of the period: T 1 = 3. Repeat the experient with a different initial deflection angle, say half of the one you used in Q. Record the value of the period T = 4. How does T copare to T 1? Explain. 5. Copute the acceleration of gravity Clearly explain how you obtained it and provide your derivations. g= 008 Penn State University Physics 11: Lab Oscillations. Siple Haronic Motion.

4 6. Copare the result above with the theoretical value and copute the percent difference below. 7. Decrease the length of the pendulu twice and easure the new period of oscillations. Is the value consistent with the theory? Explain. Lab Activity II: Spring Oscillator Goals: Deterine the period and angular frequency of a spring oscillator and copare their values with theoretical prediction. Check how the period depends on the aplitude of the oscillations. Check how the period depends on the ass of the cart and the spring constant. Investigate how kinetic and potential energies depend on tie and check conservation of total echanical energy. Setting Up the Graphs: Set up Data Studio to read the data collected by the otion sensor located at the base of the track. Check to ake sure that the otion sensor is oriented towards the cart and is sending out a narrow bea signal. Create a graphing window that contains the three following graphs: Position vs. Tie, Velocity vs. Tie, and Acceleration vs. Tie. Using the Experient Calculator, create new calculation called Position fro Equlibriu. Based on that, create calculations Potential Energy of the Spring, Kinetic Energy and Total Mechanical Energy. 1. Measure the ass of the cart and record the value. =. Deterine the spring constant of each spring. Explain how you did it and record the values: k 1 = k = 3. Using the ruler located along the length of the track, estiate the distance fro the otion detector to the closest end of the cart when the cart is at rest in the equilibriu position. Record this value in the table below. Distance of Cart fro Motion Detector at Equilibriu (eters) 4. Stretch the spring soe initial length (but do not over-stretch the spring!) and then release the cart fro rest. Press Record to collect data for approxiately five cycles of siple haronic otion, and then press Stop. Insert the graphs of x(, v(, and a(. 008 Penn State University Physics 11: Lab Oscillations. Siple Haronic Motion.

5 5. How are these graphs related to one another? What is the location and direction of the cart when the velocity is at a axiu or iniu value? What is the location of the cart when the velocity is equal to zero? When the acceleration of the cart is at a axiu or iniu value, what is the velocity and location of the cart? Deterine the Angular Frequency of the Syste: 6. Using a ethod of your choice, easure the aount of tie that it takes the cart to coplete 1 cycle. 7. Fro your easureent, record and/or calculate the period, frequency, and angular frequency of the resulting siple haronic otion of the cart/spring syste in the table below. Period T (s) Frequency f (Hz) Angular Frequency ω (rad/s) 8. Deterine the theoretical value for the angular frequency, ω, of the cart and spring syste. (Theoretical) Angular Frequency - ω, (rad/s) 9. What is the value of the % error between the experiental value of ω and the theoretical value of ω (assued to be true )? Energy: 10. Insert the three graphs with energy below. 11. What is the period of the Kinetic Energy? Potential Energy? 1. Is total Energy constant over tie? Explain. Checking the dependence of the period on the paraeters of the syste: 13. Change the ass of the cart and deterine the new period of oscillations. Is it consistent with the theory? 14. Change the spring and deterine the new period of oscillations. Is it consistent with the theory? 15. Asseble an inclined spring oscillator with the sae spring constant and ass of the cart. How does the new period copare to the horizontal oscillator? 16. Asseble a vertical spring oscillator with the sae spring constant and ass of the cart. How does the new period copare to the horizontal oscillator? 008 Penn State University Physics 11: Lab Oscillations. Siple Haronic Motion.