Physics Lab - Simple Harmonic Motion

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1 Simple Harmonic Motion Lab Online Purpose The purpose of this lab is to study some of the basic properties of Simple Harmonic Motion (SHM) by examining the behavior of a mass oscillating on a spring. Theory One type of motion is called periodic motion. In this type of motion, the behavior, called the cycle, is repeated again, again, and again over a particular time interval, AKA a period. For periodic motion, the mass will always follow the same path and return to its original location at the end of each cycle. In an ideal system this behavior would go on forever, but in reality, it goes on till the mass losses all its mechanical energy. All periodic motion has some basic properties in common. Those properties are: 1. The Cycle - The motion that is being repeated. 2. The Amplitude ( 𝐴 ) – The magnitude of the mass’s furthest displacement from its equilibrium position during the cycle. 3. The Period ( 𝑇 ) – The time it takes to complete one cycle. 4. The Frequency ( 𝑓 ) -- The number of cycles completed per unit time. (The frequency is the mathematical inverse of the period.) 𝑓 = 1 𝑇 5. The Angular Frequency ( 𝜔 ) – The frequency multiplied by 2π. 𝜔 = 2𝜋𝑓 = 2𝜋 𝑇 One particular subcategory of periodic motion is Simple Harmonic Motion (SHM). SHM has two more properties: 1. The restoring force acting on the mass must be proportional to the displacement of the mass from its equilibrium position, and pointing in the opposite direction of the displacement. 𝐹 ௦ = −𝑘 ∙ ∆𝑥 (The equilibrium position then is, by definition, the location where there is no restoring force acting on the mass. 𝑘 is the Spring Constant of the device applying the resorting force.) 2. The period of oscillation is independent of the value of the amplitude of oscillation. As an example, for an oscillator to be a SHM oscillator, it doesn’t matter if its amplitude is set to be 10 cm or 10 km, once set in motion the time it takes for that oscillator to complete one cycle MUST BE THE SAME.

2 We know from Newton’s Second Law that all forces can be written as 𝐹 = 𝑚𝑎 so we can set the standard force equation equal to the restoring force and see that: 𝑎 = − 𝑘 𝑚 ∙ ∆𝑥 In the particular case of a mass attached to an ideal spring, the frequency of oscillation will be related to the mass and the force constant by: 𝑓 = 1 2𝜋 ඨ 𝑘 𝑚 And therefore as well: 𝑇 = 2𝜋ට 𝑚 𝑘 𝜔 = ඨ 𝑘 𝑚 It can therefore easily be shown that the magnitude of the maximum acceleration the SHM oscillator will experience during a cycle is given by: 𝑎 ௠௔௫ = 𝜔 ଶ ∙ 𝐴 Also, using the fact that linear speed is related to angular speed by 𝑣 = 𝜔 ∙ 𝑟 , we can see that the magnitude of the maximum speed the SHM oscillator will experience during a cycle is given by: 𝑣 ௠௔௫ = 𝜔 ∙ 𝐴

3 Setup: Measuring the Spring Constant 1. Go to the following website: https://phet.colorado.edu/en/simulation/masses-and-springs 2. You should now see the following: 3. Click on “Download”, and then open the software when it has completed downloading 4. You should now see the following:

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4 5. Double click on “Lab” that you see on the right side of your screen. 6. Now you should see the following. 7. In the grey box on the right side of your screen make sure the following are selected. a. Displacement and Natural Length b. Movable line c. Gravity is set to 9.8 m/s 2 (Earth) d. Damping is set to “None”. 8. From the small grey box near the bottom of the right side of your screen, drag the ruler out and place it so its zero slash is aligned with the Natural Length line that should now be at the bottom of the vertical spring. 9. In the white box at the top and slightly to the right of your screen, set the Spring Constant on space to the first setting to the right of “Small”. 10. Your screen should now look like this.

5 11. Please note where the Natural Length Line is touching the spring. This is the location on the spring that you will be measuring the displacement in reference to as you add mass to the spring. Procedure: Measuring the Spring Constant 1. By clicking on and dragging, attach the Orange mass near the left and bottom of your screen to the spring. 2. In the white box near the top and right of your screen set the mass to 50 grams. a. Record this mass in Kg, in Table 1. b. Click on the little red stop sign near the top center of your screen to make the spring stop oscillating. 3. Now move the Movable Line so the red dashed line touches the reference location on the spring. 4. Use the ruler to measure the displacement between the Natural Length Line and the Movable Line. Record this displacement in Table 1. a. Repeat this process for the following masses, 75 g, 100 g, 125 g, 150 g, 175 g, 200 g, 225 g, 250 g, 275 g, and 300 g. Set up: Measuring the Frequency of Oscillation 1. Click on the yellow button at the bottom right of the screen to reset the simulator. 2. In the grey box at the left of your screen, make the following settings. a. Set Gravity to 9.8 m/s 2 , Earth b. Set Damping to none 3. In the small grey box near the bottom right of the screen, click and drag the clock and place it somewhere on the right of your screen. 4. In the white box at the top right of your screen, reset the Spring Constant to the first setting to the right of “Small”. This is the same setting that was used in the first part of the experiment. 5. In the white box at the top left of your screen set the Mass to 100 g. 6. Your screen should now look like the following.

6 Procedure: Measuring the Frequency of Oscillation 1. Click and Drag the Orange mass (0.100 kg) and attach it to the spring. This will cause the spring to start oscillating. a. Record this mass in Table 2. 2. You are going to want to measure the time it takes for the spring to complete 10 oscillations. a. One oscillation is the spring starting at its max height, going to minimum height, and then returning to its max height. b. Using the clock, measure the time it takes the spring to complete 10 oscillations, and record that time in Table 2.

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7 Analysis of Simple Harmonic Motion Lab Online Name Melissa Fernandez Ayala Group# NA Course/Section PHY 1951 011 Instructor Christopher Dunn, TA: Amilcar Torres Quijano Table 1 (20 points for first and third columns) Mass (kg) Weight (N) Displacement, Δy (m) 0.05 kg 0.49 0.130 m 0.075 kg 0.74 0.185 m 0.100 kg 0.98 0.250 m 0.125 kg 1.23 0.315 m 0.150 kg 1.47 0.375 m 0.175 kg 1.72 0.435 m 0.200 kg 1.96 0.500 m 0.225 kg 2.21 0.560 m 0.250 kg 2.45 0.620 m 0.275 kg 2.70 0.680 m 0.300 kg 2.94 0.740 m 1. Calculate the weight for each mass and fill in the middle column. Show the work for one of these calculations to receive credit. (5 points) W (N) = mg G = 9.81 ௠ ௦^ଶ Mass: 50 g x 10 -3 = 0.050 kg 0.050 kg x 9.81 ௠ ௦^ଶ = 0.49 N Mass: 75 g x 10 -3 = 0.075 kg 0.075 kg x 9.81 ௠ ௦^ଶ = 0.74 N Mass: 100 g x 10 -3 = 0.100 kg 0.100 kg x 9.81 ௠ ௦^ଶ = 0.98 N Mass: 150 g x 10 -3 = 0.150 kg 0.150 kg x 9.81 ௠ ௦^ଶ = 1.47 N Mass: 175 g x 10 -3 = 0.175 kg 0.175 kg x 9.81 ௠ ௦^ଶ = 1.72 N

8 Mass: 200 g x 10 -3 = 0.200 kg 0.200 kg x 9.81 ௠ ௦^ଶ = 1.96 N Mass: 225 g x 10 -3 = 0.225 kg 0.225 kg x 9.81 ௠ ௦^ଶ = 2.21 N Mass: 250 g x 10 -3 = 0.250 kg 0.250 kg x 9.81 ௠ ௦^ଶ = 2.45 N Mass: 275 g x 10 -3 = 0.275 kg 0.275 kg x 9.81 ௠ ௦^ଶ = 2.70 N Mass: 300 g x 10 -3 = 0.300 kg 0.300 kg x 9.81 ௠ ௦^ଶ = 2.94 N 2. Using Excel or some other graphing program, plot Weight vs Displacement . Use the trendline to find the slope of your graph . The slope is the Spring Constant of this spring. Record the value of the slope in the space provided below. Turn in this graph with the lab worksheet. Make sure the trendline is displayed on the graph. Measured Spring Constant, k (N/m): 3.99 (10 points)

9 Table 2 (5 points) Mass (kg) 0.100 kg Time (s) 9.76 s Calculate the experimental value of the period of oscillation by simply dividing the time you measured by 10. Average Period, T avg (s) : 9.76s / 10 = 0.976 s 1. Using 0.100 kg as the mass, and the value of your experimental force constant, calculate the theoretical value of the period for your spring mass system . (5 points) T th = 2π ට ௠ ௄  2π ට ଴.ଵ଴଴ ௞௚ ଷ.ଽଽ  0.995 s m = 0.100 kg K = 3.99 2. Calculate the % error between your experimental and calculated period. (5 points) | ௩௔ି௩ ௩௘ | x 100%  | ଴.ଽ଻଺ ି଴.ଽଽହ ଴.ଽଽହ | x 100%  1.91% 3. In theory, relative to the equilibrium position, where is the mass when its speed is at its maximum ? (10 points) The speed of the mass is at its maximum when the mass is at the equilibrium position, as depicted in the image. Here, the displacement from the equilibrium is at its maximum, causing the restoring force to be at its peak and driving the mass to move fastest through the equilibrium position.

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10 4. In theory, relative to the equilibrium position, where is the mass when its speed is zero ? (10 points) When the speed of the mass is zero, it is at its maximum displacement from the equilibrium position (maximum amplitude). This essentially occurs at the extremes of the motion, which are when the mass reaches the furthest point from the equilibrium on either side (shown in the image above). At these points, all the energy has been converted into potential energy, temporarily bringing the motion to a stop before the direction of the motion reverses. 5. In theory, relative to the equilibrium position, where is the mass when its acceleration is at its maximum ? (10 points) When the acceleration is at its maximum, the mass of the object is at its maximum displacement. This occurs when the mass is passing through the equilibrium position, experiencing the maximum restoring force. At this point, the acceleration is at its peak due to the strong force acting to bring the mass back towards the equilibrium position. 6. In theory, relative to the equilibrium position, where is the mass when its acceleration is zero ? (10 points) When the acceleration is zero, the mass of the object is at its equilibrium position. At this point, the restoring force is at its minimum, resulting in the mass experiencing no acceleration. In other words, the equilibrium position is where the force acting on the mass is balanced, causing the acceleration to be zero.

11 7. Do the results of our experiment confirm theoretical predictions? Explain your answer. (10 points) Considering I got a percent error lower than 10%, 1.91%, the results of our experiment do confirm my theoretical predictions. In addition, the experiment itself does confirm the theoretical predictions within the limits of experimental errors. Although not perfect, the mass (spring) was seen to move in a periodic motion by accelerating as it moved toward the equilibrium point then decelerating as it moved away from the equilibrium point.

Physics Lab - Simple Harmonic Motion

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