Lab 6-SHM - PHY 207-lab6 PDF

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PHY 207-lab6...

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Physics 207

Lab #6- Simple Harmonic Oscillators Introduction The objective of this lab was to study the motion of a mass hanging on a spring. This motion is known as Simple Harmonic Motion. We were able to use Logger Pro to detect the motion as well as the height that the mass attached to the spring reached. Through this experiment we were able to verify the basic relationship between the mass and the motion of the mass attached to the spring. When doing the experiment, we had multiple variables including the amplitude and the mass. Changing the mass and the amplitude would not give us a certain proof as to which one has an effect on the motion of the spring and which does not. Therefore, we tested for the effect of mass by using different masses and keeping the amplitude constant. Then we checked for the effect of the amplitude by changing the amplitude and keeping the mass constant. This experiment is important to make verify, experimentally, that the basic relationship that we assume in our calculations are accurate, and know how the mass, the length of the spring, and the amplitude affects the motion. We were able to find the experimental value to be within the level of uncertainty of our experiment from the accepted value. Make a sketch of the experimental setup to have in your report. Procedure First, we needed to find the value of spring constant, k. We used the equation F=-kx to calculate the spring constant. In order to come up with the value of x we measured the length of the spring without any weights on it. Then, we measured the length of the spring after we attached a 100g mas to it, and the difference between the stretched and unstretched spring is the value of x. Knowing that the only force acting on the mass is the force of gravity, we were able to calculate the value of k.

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After that and before we take any records, we sketched the graph that we expected to be the motion of the spring when it is oscillating at an amplitude A. Then, we connected the Logger Pro to record the motion of the spring, and through that we were able to collect the graphs for position vs. time, velocity vs. time, and acceleration vs. time. Then, we tested for what affects the motion of the spring which is the purpose of this experiment. First, we tested for the affect of change of mass on the motion of the spring. We added a mass of 0.2kg and pulled the spring to an amplitude, A, and released the spring. We recorded the period of the spring to make one full oscillation and recorded it to the data sheet. Then, we repeated the same steps but by changing the mass to 0.5kg and kept the same amplitude, A, and recorded the period of the spring and recorded it to the data sheet. Finally, we added a mass of 1 kg to the spring to collect another set of data. We pulled the spring to the same amplitude, A, and recorded the period of the spring. Through the set of data collected for the three different masses we were able to sketch a graph and find if mass has an effect on the period of the spring. Also, using the same set of data we compared the period this time with the square root of the mass to find if it has a relationship with the period. In addition, we sketched a graph to find if the period depends on the square root of mass. Following, we found if the change in the amplitude affects the period of the spring. Therefore, we chose a mass of 0.5kg and kept it constant throughout this experiment. However, we changed the amplitude. First, we pulled the spring to an amplitude of 0.012m and released it. after we released it, we recorded the period, which is the time it took to make a full oscillation. Then, we kept the 0.5kg but this time we changed the amplitude to 0.007m and recorded the period. Finally, we changed the amplitude to 0.018 and recorded the period of the spring. After

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collecting the data, we plotted it into a graph to see if the change in period depends on the amplitude. Finally, we calculated the energy before and during the motion of the spring to check for the concept of conservation of energy. First, we recorded a set of data for mass of 1kg using Logger Pro. Then, we found the energy before the spring was released and when it reached an amplitude of zero. Before the spring was released all the energy was in form of potential energy. Therefore we used the equation U=(0.5)*(k)*(x2) to find the initial energy. Then, at the amplitude qual zero all the energy was in form of kinetic energy. Therefore, we used the equation K=(0.5)*(m)*(v2) to find the final energy. The velocity we obtained from the graph recorded by Logger Pro, in which the velocity at amplitude equal zero was the maximum velocity. Data and graphs The value of spring constant k= 38.9 N/m Energy at the highest position = 0.14 J Energy at the equilibrium position = 0.1405 J Prediction for position vs. time (d vs. t), velocity vs. time (v vs. t) and acceleration vs. time (a vs. t) graphs for oscillation that lasts for 2 periods.

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M = 200 g 1 0.5 0 -0.5.05 .25 .45 .65 .85 .05 .25 .45 .65 .85 .05 .25 .45 .65 .85 .05 .25 .45 .65 .85 .05 -10 0 0 0 0 1 1 1 1 1 2 2 2 2 2 3 3 3 3 3 4 -1.5 Time (seconds)

Position Velocity Acceleration

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Mass = 500 g 6 4 2 0 -2.05 0.3 .55 0.8 .05 1.3 .55 1.8 .05 2.3 .55 2.8 .05 3.3 .55 3.8 .05 0 0 1 1 2 2 3 3 4 -4

Position Velocity Acceleration

-6 Time (seconds)

Positon, Velocity, Acceleraton

Mass = 1000 g 4 3 2 1 0 -1.05 .25 .45 .65 .85 .05 .25 .45 .65 .85 .05 .25 .45 .65 .85 .05 .25 .45 .65 .85 .05 -20 0 0 0 0 1 1 1 1 1 2 2 2 2 2 3 3 3 3 3 4 -3 -4

Position Velocity Acceleration

Time (seconds)

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Sqrt(Mass) (kg) vs Time (s) 1.2

Time (s)

1

f(x) = x

0.8 0.6 0.4 0.2 0 0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

1.05

1.1

1.15

Sqrt(Mass) (kg)

√ Mass ( g) ∧Time(s)

Relationship between

Time (s)

Amplitude (m) vs Time (s) 0.51 0.5 0.49 0.48 0.47 0.46 0.45 0.44 0.43 0.42 0.1

f(x) = − 0.82 x + 0.59

0.11

0.11

0.12

0.12

0.13

0.13

0.14

0.14

Amplitude (m)

Sample calculations 

To find the spring constant, k, use the following equation, F=-kx. (1)(9.8)=(0.252)k, k=38.9 N/m



To find the potential energy at the highest point use the following equation, U=(0.5)(k) (x2). U=(0.5)(38.9)(0.085)2=0.14 J.



To find the kinetic energy at the point of the equilibrium use the following equation, K=(0.5)(m)(v2). K=(0.5)(1)(0.53)2=0.1405 J

Questions 1. Describe how the mass moves relative to the equilibrium position. 6

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The mass moves relative to the equilibrium position by oscillating up and down. What causes the oscillation the mass on the spring is the force of the spring. 2. Calculate the maximum velocity from the position vs. time graph. Show your calculations. Compare with the value from the velocity time graph. v=

x 2−x 1 0.283889−0.245641 0.038248 = = =0.55 m /s 1.17−1.10 t 2 −t 1 .07

The value the we obtained through the graph was 0.53 m/s. 3. At what position is the velocity a maximum? The accurate position of the maximum velocity is at the equilibrium position. However, in this lab it was at x=0.07 due to the sources of error that contributed to the results due to air resistance that we did not include in our calculations for this lab. 4. Calculate the minimum velocity from the position vs. time graph. Show your calculations. Compare with the value from the velocity time graph. v=

x 2−x 1 0.345378−0.354764 0.009386 = =0.05 m/ s = 0.85−0.80 .05 t 2 −t 1

The value that we obtained from the graph was 0.056 m/s. 5. At what position is the velocity a minimum? The velocity is minimum at the position x=0.06. This distance is the distance away from the top and bottom of the oscillation. 6. Calculate the maximum acceleration from the velocity vs. time graph. Show your calculations. Compare with the value from the acceleration vs. time graph. a=

v 1+1−v 1 −0.0497105 −0.2840032 −0.3337137 = =−6.674274 m / s2 = t 1+1−t 1 .05 0.85−0.80

The value that we obtained from the acceleration vs. time graph was -7 m/s 2 .

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7. At what position is the acceleration a maximum? The position in which the acceleration is maximum is at x=0.08. 8. Calculate the minimum acceleration from the velocity vs. time graph. Show your calculations. Compare with the value from the acceleration vs. time graph. a=

v 2−v1 −0.8128414−−0.7961716 −−0.0166698 =−0.23814 m/s 2 . = = .07 1.17−1.10 t 2 −t 1

The value obtained from the graph was 0.207 m /s 2 . 9. At what position is the acceleration a minimum? Acceleration is minimum at the position x=0.2m. 10. Compare your position, velocity and acceleration graphs with your predictions on page 1. Resolve any discrepancies.

Time (s)

Amplitude (m) vs Time (s) 0.51 0.5 0.49 0.48 0.47 0.46 0.45 0.44 0.43 0.42 0.1

f(x) = − 0.82 x + 0.59

0.11

0.11

0.12

0.12

0.13

0.13

0.14

0.14

Amplitude (m)

The plots generally match our expectations, but a more expected result would have been if all the points were aligned. That would have been possible if the measurements were taken with more accuracy.

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11. Use the plots mentioned above (T(m), T(sqrt(m)), T(A)) to validate the relations between the period and the mass or amplitude. There is a direct relationship between the period and the square root of mass. As the square root of mass increases the period of the spring increases linearly. 12. What is the time constant, , of this damped oscillator? y=−0.0020 x−0.0072 A (t ) y=ln[ ] A

t=x

m=

−1 t

−1 t t=500 seconds −0.0020=

13. Does the period change as the system loses energy? Explain. The period does not change as the system loses energy. However, the amplitude changes because the period depends on mass, and even though energy is being lost, the mass remains constant. 14. The reduction in amplitude represents a loss of energy by the system. Where does the energy go? Loss of energy is always released to the atmosphere. The energy goes into the heating of the air (by air resistance) and the internal heating of the spring as its crystals slide past each other. This internal heating is not noticed however because the heat will be quickly dissipated into the surrounding air. Conclusion The result of this experiment was expected. Also, the result matched the predicted graph for the motion of the spring. At the end of the experiment we found the energy at the highest point almost equal to the energy at the point of equilibrium, inly 0.0005 J difference which is

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within the level of uncertainty for this experiment. Many factors affected the outcome of our experiment. One of those factors was the condition of the instrument. The pole which the spring and the mass was attached to was slightly moving, which affected our measurements, calculations, and therefore the results of the experiment. Measurements also affected the accuracy of the experiment’s result. We were measuring the distance in which the spring was stretched after we added the mass to it using a meter stick. Therefore, there could have been slightly different value for the distance which could contribute to the result. Also, we assumed that the amplitude remained the same in one of the experiments. However, due to human error, the person holding the mass could have, unintentionally, changed the amplitude slightly. To improve these factors next time, we could use instruments that are in a better condition. Therefore, the pole would not interfere with our measurements our or our results. Also, we should double check on the measurements to avoid errors. Also, we should use pen to mark the amplitude, so we make sure that we start at the same amplitude every time and that does not contribute as a source of error. By improving those sources of error, we would be able to get a more accurate result. In the experiment, the objective was met successfully. We learned how does the square root of mass affect the period, and how the amplitude does not have an effect.

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