CALIFORNIA STATE UNIVERSITY, LOS ANGELES

Department of Physics and AstronomyPhysics 212-14 / Section 14- 34514Masses on Springs: Simple Harmonic MotionPrepared by: Faustino Corona, Noe Rodriguez, Rodney Pujada, Richard LamPerformance Date: Tuesday,March 30, 2016Submission Due: Tuesday, April 6, 2016Professor: Ryan AndersenWednesday: 6:00 pm. 8:30 p.m.April 2016Experiment No 1: Masses on Springs: Simple Harmonic MotionI. ABSTRACT:To study how the simple harmonic motion work by analyzing its repetitive motion and determine its mathematical solution by using a spring.II. INTRODUCTION. When we look at a spring under some sort of tension, the force of the spring can be applied to this formula which k represents the springs constant.

F= -kxThis formula is called Hookes Law. Hookes Law states when a spring is stretch, it is directly proportional to the force that is doing so. The graph as long as the spring can return to its original shape without being deform. The formula is relevant. Another to write the formula is.

mg=ksIn Newtons 2nd law, the equation of motion mass is k(s-x)-mg= md2xdt2And by combining the last two formula we get

d2xdt2+kmx = 0III. EXPERIMENTAL PROCEDUREThe equipment used in this experiment is a meter stick, a spring, a set of weights ranging from 10 grams to 200 grams, and a stopwatch. The spring is hanging parallel to the meter stick in order to measure its displacement. To measure the timing uncertainty, time two periods using moderate amplitude several times. The uncertainty calculated was 0.07s.

For part A, hang a weight from the spring and time 10 periods for small, medium, and large amplitudes. These measurements will determine the period for each amplitude. For Part B, use 5 different masses and measure the period for each mass using the same technique used in part A. For part C, use a scale to weigh the masses used and the mass of the spring. Finally, for part D, hang a 50 gram weight to the spring and measure its displacement. Our reading uncertainty for displacement measurements was 1.0cm. Gradually add more weight and measure the displacement at equilibrium for each weight. Continue to add weights and collect measurements until the spring is stretched to 100 times the reading uncertainty.IV. DATA AND ANALYSIS

4.1. CALCULATION OF THE PERIOD OF OSCILLATION

DATA TABLE 1

ItemValue (second)

Period 11.73

Period 22.37

Period 32.65

Period 42.6

Period 52.22

Average period of oscillation2.31

DATA TABLE 2

ItemValue

Mass of the spring0.0335 kg

Effective mass of the spring (one-third mass of the spring)0.026 kg

Mass of the hanging mass0.500 kg

Mass, m (hooked mass plus effective mass of the spring)0.526 kg

4.2. DETERMINE THE PERIOD FOR SMALL, MEDIUM AND LARGE AMPLITUDESDATA TABLE 4

Displacement (cm)Displacement (m)T (seconds)No CyclesPeriod (T)

10.0110.61101.061

50.0510.62101.062

120.1210.69101.069

Graph Period vs displacementDisplacement (m)Period (T)

0.011.061

0.051.062

0.121.069

Graph No 1 Period vs displacement

4.3. Graph T2 Vs MassDATA TABLE 5ItemTime (second)Mass ( Kg)

T16.940.05

T212.40.713

T311.410.1417

T36.820.0443

T410.780.2771

Average period of oscillation9.67

Amplitude = 3 cm = 0.03 mMass of spring 33.5 g

DATA TABLE 6 :

T (seconds)No CyclesPeriod (T)Mass (grams)Mass (Kg)

6.94100.69450.00.050

12.4101.24173.00.173

11.41101.141141.70.142

6.82100.68244.30.044

10.28101.078127.10.127

If the mass is pulled so that the spring is stretched beyond its equilibrium (resting) position, the restoring force of the spring will cause an acceleration back toward the equilibrium position of the spring, and the mass will oscillate in simple harmonic motion. The period of vibration, T, is defined as the amount of time it takes for one complete oscillation, and for the system described above is:

where me is the equivalent mass of the system, that is, the sum of the mass, m, which hangs from the spring and the spring's equivalent mass, me-spring, or

Note that me-spring is not the actual mass of the spring, but is the equivalent mass of the spring. It is not the actual mass because not all of the mass pulls down to act in concert with the weight pulling down. Its theoretical value for our system should be approximately 1/3 of the actual mass of the spring. Substituting equation 4 into equation 3 and squaring both sides of the equation yields:

Therefore, if we perform an experiment in which the mass hanging at the end of the spring (the independent variable) is varied and measure the period squared (T2 ; the dependent variable), we can plot the data and fit it linearly. Comparing equation 5 to the equation for a straight line (y = mx + b), we see that the slope and y-intercept, respectively, of the linear fit is:

We calculate T2 and graphingPeriod (T)Mass (Kg)T2

0.6820.0440.465

0.6940.0500.482

1.0780.1271.162

1.1410.1421.302

1.240.1731.538

Where the slope is 8.5445 . Therefore 8.5445 = 4 2 / K

K = 4.62 Newton/meter

4.4. DETERMINE K CONSTANTDATA TABLE 7Mass (kg)X ( cm )X (m)

0.0570.50.705

0.0672.50.725

0.0774.50.745

0.0876.50.765

0.0978.50.785

0.1081.50.81

0.12850.85

0.1591.50.915

0.201020.102

0.251130.113

0.30123.50.1235

0.40144.50.1445

5.00176.50.1765

Amplitude = 3 cm = 0.03 mMass of spring 33.5 g

F = kx

f = 0

Mg = Kx

Therefore: K = mg / X

For mass 0.05 Kg We multiply by 9.8 m/s2 : (0.05 Kg) (9.8 m/s2 ) = 0.49 N

Graph No 3: Determine K constantX (m)mg ( Newton)

0.020.588

0.040.686

0.060.784

0.080.882

0.110.98

0.1451.176

0.211.47

0.3151.96

0.4252.45

0.532.94

0.743.92

1.064.9

By the graph we determine K = 4.2971 Newton/meter

K = 4.2971 Newton/meter

4.5 Calculate the percent of errorWe appreciate the part 2 by the graph T2 vs mass and the calculation for K constant has more precision in their measurement and our calculation shows less and less uncertainty for each experiment..

Calculate the percent of error:

Percent error = ( Dpractical Dtheoric) x 100 % formula 1 D theoric

Data:K= 4.62 Newton/meter ( graph T2 vs mass)K = 4.2971 Newton/meter ( by the graph force vs mass)

Using the formula No 1 to evaluate percent of error.

Percent error = (4.62 - 4.30) x 100 % = 7.44 %

4.30Percent error = 7.44 %

V. RESULTSThis experiment has three parts: Our first experiment show us the period of the spring was measured as amplitude changed while mass remained constant. The period remained nearly the same throughout every trial, which was to be expected. Any differences in period may be accounted to inadequate stopwatch usage and inaccurate starting displacements throughout the trials that can appreciate in Graph No 1.

In our second part we calculate K constant by Graph T2 Vs Mass that make a K = 4.62 Newton/meter by the slope slope = 42/K.

During part three of the experiment, the vertical displacement of a spring was measured as a function of force applied to it. Mass was added to the spring, and the displacement was recorded. This was repeated with various amounts of mass. From these data, a graph of force versus displacement was plotted, and a linear fit slope revealed the spring constant. In this endeavor, the spring constant was valued at 4.30 N/m.VI. CONCLUSIONS

From the results of our experiment we can observe that in the insistence of Period (Time) vs. Amplitude we notice that no matter if you stretch it 1cm or 12cm the Period would be fairly similar. This dependence since timing is a big factor when doing this experiment thats why I said fairly similar because of the time issue and measurements. But when you observe the experiment when amplitude is no longer in account but weight (Mass) is you would notice the change in time with the more or less weight on it. If less weight is add the less time it would take to make the ten cycles we are using to observe this experiment, but if you add more weight the longer the time would be to reach its ten cycles we are observing from. Some agreements with the predictions is that the amplitude should not have an effect on period which it does not and that weight does have an effect on period depending on the amount of weight placed on the spring. I have to disagree with some results since without precision some numbers would be off. For example period because timing is an issue since we can never get the time right this is why we had to do ten cycles to get the most precision in this experiment. Another issue was measurements because we did not have a precise measurement technique we had to eyeball where it started to were its amplitude was at. Meaning that in some instances the amplitude we want to get it to can be a bit over or under meaning that this can make the error a bit more than it should. Some possible improvements are finding new ways to measure distances more accurately and time as well. VII. REFERENCES Department of Physics and Astronomy CSU Los Angeles. Edition 2.0, XanEdu Custom Publishing, pp. 8-14

VIII. DATA SHEETS