Simple Harmonic motion PDF

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Class: PHY 215 Prof. Mahmoud Ardebili SIMPLE HARMONIC MOTION Objective: To explain why simple harmonic motion is simple and harmonic, and to better understand how the period of mass oscillating on a spring varies with mass and the spring constant. Introduction: The restoring force F exerted by a stretched spring is proportional to the stretching distance x. Equation, F = -k*x, is known as Hook’s Law, where x is distance and k is spring constant. The constant k is related indication of the stiffness of the spring. Under the influence of a linear restoring force described by Hooke’s Law, a particle or object undergoes that known as Simple Harmonic Motion. This periodic oscillatory motion can be found very common in the nature. In this experiment, description of simple harmonic motion and Hooke’s Law will be investigated. Theory: A. Hooke’s Law: Hooke’s Law is expressed as F = -k Δ x = -k (x - xo) where xo = 0. The minus sign represents that the force and displacement are in opposite direction. The k is spring constant. According to Hooke’s Law, the elongation of a spring constant is directly proportional to the stretching force. If a spring has an initial length y0, and a suspended weight of mass m stretches the spring to a length y1, and the spring force is balanced as F1 = mg = k (y1 – y0). Here y represents the length in the vertical direction. If another mass is added and spring stretched to another length y2, then same formula will be use but with different numbers. B. Simple Harmonic Motion: Periodic motion is when the motion of an object is repeated in regular time. The reason behind calling Simple Harmonic motion is the simple means restoring the force in simplest form and harmonic means that it can be described by sines and cosines function. The period of oscillations performed in the experiment helps to determine the k. The period of oscillation depends on the parameters like mass of a spring and represents T = 2π√(m/k). If the equation is manipulated, from T vs, mass graph it can find slope and the other equation can be represented as k = (2π)2 / slope. Equipment Needed: Using Website phet.colorado.net  Different masses hanger  Scale  Stopwatch  Spring Procedure: Spring Elongation: 1) Go the website phet.colorado.net, select Masses and Springs and select the Lab part. 2) Now set the scale provided on the right side along the spring bottom tip and set the spring constant size small at first.

3) Start with attaching the weight of 50 g to the spring and record the scale reading in data table. Now add another 50 g weight and record the scale reading in the data table. Repeat these up to 6 times in total. 4) Next, set the spring constant size large and repeat the procedure of step (3). 5) In excel, draw the graph and calculate the slope using slope formula in excel for both data tables. Calculate the k for two different spring constant size using formula k = (2 π) 2 / slope. Take the average value of k and record it. Period of Oscillation: 6) Keep the same set up of scale and this time set up stopwatch provided on the right side. 7) Hang the mass of 100 g to the spring and pulled the hanger of mass by 10 to 12 cm by looking at the scale. 8) After pulling the hanger, start the stopwatch and determine the time of 5 to 7 oscillations respected to that mass. Record the data and calculate the average time. 9) Repeat the procedure (7) and (8) for the different masses like 150 g, 200 g, 250 g, and 300 g. 10) Plot a graph of the average time square and the mass and, calculate the slope in excel. 11) After finding the slope, calculate the k by using formula k = (2 π)2 / slope. 12) At the end, calculate the percent difference between the k of spring elongation and k of period of oscillation. DATA TABLE 1: SPRING CONSTANT WITH SMALL SIZE M 1g M 2g M 3g M 4g M 5g M 6g

Total Weight (mass in kg) g N (0.050 kg) g N (0.100 kg) g N (0.150 kg) g N (0.200 kg) g N (0.250 kg) g N (0.300 kg) g N

Y1 Y2 Y3 Y4 Y5 Y6

Scale Reading (m) 0.08 0.16 0.24 0.36 0.49 0.57

Total Weight Vs. Scale Reading (Small Spring Constant Size) 0.6

Scale Reading (m)

0.5 0.4 0.3 0.2 0.1 0

0

0.05

0.1

0.15

0.2

Total Weight (mass in kg) g N

0.25

0.3

0.35

Slope: Calculated using slope formula [=SLOPE (A2:A7, B2:B7)] in excel: 0.48 k = (2 π)2 / slope = (2 π)2 / 0.48 = 82.25 N/m. DATA TABLE 2: SPRING CONSTANT WITH LARGE SIZE M 1g M 2g M 3g M 4g M 5g M 6g

Total Weight (mass in kg) g N (0.050 kg) g N (0.100 kg) g N (0.150 kg) g N (0.200 kg) g N (0.250 kg) g N (0.300 kg) g N

Y1 Y2 Y3 Y4 Y5 Y6

Scale Reading (m) 0.04 0.09 0.12 0.17 0.24 0.29

Total Weight Vs.Scale Reading (Large Spring Constant Size) 0.35

Scale Reading (m)

0.3 0.25 0.2 0.15 0.1 0.05 0 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Toatel Weight (mass in kg) g N

Slope: Calculated using slope formula [=SLOPE (H2:H7, I2:I7)] in excel: 0.98 k = (2 π)2 / slope = (2 π)2 / 0.98 = 40.28 N/m. Average value of k of spring elongation: [82.25+40.28]/2=61.27 N/m. DATA TABLE 3: PERIOD OF OSCILLATION Total Mass Suspended (mass in kg) g N (0.100 kg) g N (0.150 kg) g N (0.200 kg) g N (0.250 kg) g N (0.300 kg) g N

Total Time in (s) 3.49 3.72 3.85 3.94 4.03

No. of Oscillation 5 5 5 5 5

Average Time in (s) 0.698 0.744 0.770 0.788 0.806

T square (T2) in (s2) 0.487 0.553 0.592 0.621 0.649

Total Weight Vs. Time Square 0.7 f(x) = 0.78 x + 0.42

0.6

Time Square (s^2)

0.5 0.4 0.3 0.2 0.1 0 0.05

0.1

0.15

0.2

0.25

0.3

0.35

Total Weight (mass in kg) g N

Slope: Calculated in excel by graph (equation): 0.78 k = (2 π)2 / slope = (2 π)2 / 0.78 = 51.3 N/m. Percent difference in k: |51.3 – 61.3| / [(61.3 + 51.3) / 2] *100 = 17.76% Conclusion: In conclusion, the value of spring constant k and the value of k by period of oscillation was not that much close and the percent difference was little bit high. May be in counting the oscillation of different mass was taken more in number then it might be possibility of getting the value closer to the spring constant. Overall, it was a new experience of doing experiment using website and it clarified the topic very well....