Simple Harmonic Motion


 Godfrey Welch
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1 Simple Harmonic Motion Simple harmonic motion is one of the most common motions found in nature and can be observed from the microscopic vibration of atoms in a solid to rocking of a supertanker on the ocean. Simple harmonic motion is very useful in measuring time. The source of the motion is a linear restoring force. That is, if the object is displaced from its equilibrium position, then there is a force that is opposite in direction of the displacement and proportional to the magnitude of the displacement. This was first described by Robert Hooke and is now referred to as Hooke s Law. F =  k y (1) where F is the force, y is the displacement from equilibrium position, and k is the spring constant. The minus sign indicates that the direction of F and direction of y are in opposite direction. The motion of an object acting under a linear restoring force is found by substituting Eqn 1 into Newton s Second Law equation. Therefore, F =  ky = ma (2) The solution of Eqn. 2 requires calculus and will not be done here. The result is that y can be a sine or cosine function. Possible solutions for the position y, the velocity v, and the acceleration a are as follows: y = A Cos(ωt + ff) (3) v =  Aω Sin(ωt + ff) (4) a = Aω 2 Cos(ωt +f) (5) where A is the maximum displacement, and ω is the angular frequency. The phase angle φ adjust the equation for different starting conditions. If x and a from Eqns 3 or 5 are substituted into equation 2, the angular frequency can be can be found. That is w = (k/m). (6) The period of the motion T is the time for one complete cycle. Therefore, T = (2p/w) = 2 p (m/k) (7) The frequency of the motion f is the inverse of the period, so f = (1/T) = (1/2pp) (k/m) (8) The mass in Eqn 6 is the mass that is vibrating in simple harmonic motion. For a massless spring, that would just be just the mass added to the end of the spring. However, a massless spring doesn t exist in the lab. To correct for the mass of the spring, a nice calculus derivation shows that vibrating mass is equal to one third mass of the spring plus suspended mass or m = 1 3 m spring + m suspended (9) 56
2 In this experiment, the student will (1) determine the force constant of the spring, (2) show that the period is independent of the maximum displacement of the mass, (3) calculate the period for different suspended masses, and (4) fit the experimental data to the equation of motion. Equipment: 1. Lab Pro 5. mass sets 2. Ultrasonic motion Detector 6. weight hangers 3. Logger Pro software 7. ring stand, clamp, and rod 4. Spring with spring constant of 8. meter stick approximately 10 N/m Procedures: 1. Measure the mass of the spring and record its value in the Data section. Suspend the spring from the support rod and measure the distance from the table top to the bottom of the spring as shown in Fig. 1. This is the unstretched length of the spring. Record the value in the data table. 2. Add a weight hanger plus 50 grams of mass to the bottom of the spring. Record the distance from the table top to the same place on the spring as used in above procedure. Record the distance and the hanging mass (including the weight hanger) in the data table. Add 50 grams to the weight hanger and repeat the measurements. Continue for a total of 5 measurements. 3. Calculate the displacement of the spring by subtracting the distance with hanging mass from the unstretched length of the spring. Calculate the weight of the hanging mass. Plot the weight of hanging mass versus displacement. Using Excel program fit the graph to a straight line. The slope of the line is the spring constant. 4. Turn on the Lab Pro. Make sure the Ultrasonic Motion Detector is plugged into port 2 and is underneath the hanging mass. The distance between the hanging mass and the Ultrasonic Motion Detector should be 50 cm. or greater. Open the following folders on the computer labeled Logger Pro / Physics with Computers / EXP15. There will be two graphs displayed Distancs vs.time and Velocity vs. Time. To take data, start the mass in motion and then place the mouse pointer on the collect button and click. The graph should display the motion of mass for 10 seconds and shut off. 5. There can be problems in the data if the hanging mass comes too close to the Ultrasonic Motion Detector. This shows up in the distance vs. time graph by the curve becoming a horizontal line. It looks like the valley of the oscillation has been cut off. To correct this problem, raise the spring support so that the hanging mass is always 31 cm above the Ultrasonic Motion Detector. If there are spikes on the graph, Figure 1 that is caused by the mass osscilating to the right and the left and the ultrasound is missing themass completely. Make sure the supports are not oscillating and that up pull the mass straight down. A clean sinusoidal curve is needed. 6. Place 200 g on the weight hanger and let it hang 57
3 motionless. Measure the equilibrium position by clicking collect to begin data collection. After the collection stops, click the Statistics Button at the top of the screen, to determine the average distance from thedetector. Record this position as (y o ) in the data table 7. Now lift the mass upward about 5 cm and release it. The mass should oscillate along a vertical line only. Click collect. Click on the Examine Button. Place the cursor on the Distance vs. Time graph and individual data points can be read. Use this to determine the amplitude and period of the motion. Take the inverse of the period to find the frequency. Calculate the theoretical value for period and a percent error. 8. You can compare your experimental data to the sinusoidal function model using the Manual Curve Fitting feature of Logger Pro. Try it with your data. The model equation in the introduction, which is similar to the one in many textbooks, gives the displacement from equilibrium. However, your Motion Detector reports the distance from the detector. To compare the model to your data, add the equilibrium distance to the model; that is, use y=y o + A Sin(ωt+θ) (10) where y o represents the the equilibrium distance. Click on the position graph to select it. Choose Curve Fit from the Analyze Menu. Select Manual at the Fit Type and then select the Sine function from the General Equation List. Logger Pro fits the curve to the equation y = A*Sin(Bt+C) +D. Compare this equation to Eqn (10) above to match the variables: e.g., φ corresponds to C, 2πf corresponds to B and so on. 9. Adjust the values for A, B, D to reflect your values for A, φ, and y o. You can either enter the values directly in the dialog box or you can use the up and down arrows to adjust the values. The phase parameter φ is called the phase constant and is used to adjust the y value reported by the model at t = 0 so that it matches your data. Since the data collection did not necessarily begin when the mass was at the equilibrium position, f is needed to achieve a good match. The optimum value for φ will be between 0 and 2π. Find a value for f that makes the model come as close to your data as possible. Write down the equation that best matches your data. 10. Repeat procedures 69 but with a 10 cm upward displacement of the mass. 11. Put 300 g on the weight hanger and repeat procedure With 300 g on the weight hanger and a 10 cm upward displacement, repeat procedures 69. Data: A. Determination of Spring Constant 58
4 Mass of spring = Distance from table top to unstretched spring = Spring Constant (Slope of Force vs. Distance Graph) = Hanging Mass Distance to Detector Displacement Force (F = Mg) B. 200g mass with 5 cm amplitude Hanging Mass = Vibrating Mass = (Eqn. 9) Equilibrium Position (yo) = Frequency (f) = C. 200 g mass with 10 cm amplitude Frequency (f) = 59
5 D. 300g mass with 5 cm amplitude Hanging Mass = Vibrating Mass = (Eqn. 9) Equilibrium Position (yo) = Frequency (f) = E. 300g mass with 10 cm amplitude Hanging Mass = Vibrating Mass = (Eqn. 9) Equilibrium Position (yo) = Frequency (f) = 60
6 Questions: 1. By comparing results for the frequency for parts B and C and parts D and E, how does the frequency of the oscillations depend on the amplitude? 2. A massspring system undergoes simple harmonic motion with amplitude A on the horizontal frictionless plane. Does the total energy change if the mass is doubled but the amplitude is not changed? Are the kinetic and potential energies at a given point in its motion affected by the change in mass? Explain. 61
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