Lab1PHY215 Simple pendulum PDF

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Borough of Manhattan Community College Physics 215 151L[3922] 2020 Summer Term(7W1) 05/27/2020

Lab Report 1

The Scientific Method: The Simple Pendulum

Ana Barcari

Instructor: Mahmoud Ardebili

Objective: The physical principle investigated in this experiment involves the principles of applying and understanding the scientific method. This experiment will allow one to clearly distinguish what variables affect the period of a simple pendulum and how both the physical relationship and experimental can be used to calculate other information, such as the value of acceleration due to gravity. To observe how the period of a simple pendulum depends on the length of the string, the mass of the pendulum bob, the strength of gravity, and the amplitude of the swing. Observe the energy in the system in real-time, and vary the amount of friction. Introduction: A simple pendulum consists of a mass m hanging at the end of a string of length L. The period of a pendulum or any oscillatory motion is the time required for one complete cycle, that is, the time to go back and forth once. If the amplitude of motion of the swinging pendulum is small, then the pendulum behaves approximately as a simple harmonic oscillator, and the period T of the pendulum is given approximately by :

where g is the acceleration of gravity (gravity on Earth = 9.81m/s2). This expression for T becomes exact in the limit of zero amplitude motion and is less and less accurate as the amplitude of the motion becomes larger. From this expression, we can use measurements of T and L to compute g. In case of a non-linear differential equation, the period T still does not depend on m (since m does not appear in the equation of motion, it cannot appear in the solution); however, the period T does depend on the amplitude of motion. We will use the symbol θ0 for the amplitude, or maximum value, of θ . The exact solution for the period T can be written as a infinite power series in θ0 T =2 π

√

L 1 9 2θ 4θ (1+ sin + sin +…) 2 g 2 64 4

Experimental procedure and Setup: In this experiment, we used the following apparatus to illustrate the scientific method. The apparatus used in the experiment is a simulation from (https://phet.colorado.edu/en/simulation/pendulum-lab) which consist: a meterstick, timer, protractor, string, pendulum bob (it is possible to introduce different masses). Calculations of experimental period T (s) and percent errors were determined for the small-angle approximation, period dependence on mass, and period dependence on length. Also, the T2 values were calculated

for the period dependence on length. Following the experiment, a plot of L versus T2 was made. The results obtained from the experiment include the validity of the scientific method to the theoretical prediction via percent errors through the three parts and determining the value of g from the experimental data.

Data sheets: Data Table1: Investigation of the small-angle approximation:

Mass, m: 0.3kg

Pendulum length, L 0.5m

Period T(sec) Experimen tal 1.411

Theoretica l 1.418

Percent Error %

0

1.445

1.418

1.9

200

1.457

1.418

2.75

0

1.461

1.418

3.03

450

1.497

1.418

5.57

0

1.547

1.418

9.09

Angle θ 50 10

30

60

0.49

Theoretical Period T:

T =2 π

√

√

0.5 L =¿ 2 π =1.418 g 9.81

s

Percent Error Calculation Steps: 1. Subtract one value from another. The order does not matter if you are dropping the sign, but you subtract the theoretical value from the experimental value if you are keeping negative signs. This value is your "error." 2. Divide the error by the exact or ideal value (not your experimental or measured value). This will yield a decimal number. 3. Convert the decimal number into a percentage by multiplying it by 100. 4. Add a percent or % symbol to report your percent error value.

Data Table2: Period dependence on mass

Θ=100

Pendulum length, L

0.5m

Period T(sec) m (kg)

Theoretica l 1.418

Percent Error %

0.1

Experimen tal 1.437

0.2

1.433

1.418

1.05

0.5

1.420

1.418

0.14

0.8

1.431

1.418

0.92

Data Table3: Period dependence on length

Θ=100 Mass, m: 0.5kg

1.33

Period T(sec) L (m)

Theoretic al 1.236

Percent Error %

T2 (s2)

0.73

1.528

0.38

Experimen tal 1.227

0.5

1.420

1.420

0.14

2.016

0.6

1.537

1.553

1.03

2.411

0.7

1.675

1.678

0.17

2.816

0.8

1.804

1.794

0.55

3.218

Graph: L versus T2

Value of g from the experimental data (slope of the graph) =9.802 m/s2 Percent error: 0.0402%

Error Analysis The biggest source of error in the experiment is human error, along with a combination of systematic and random error. Another source of error is the measurement of the completed oscillations of the bob. An in accurate timing of the bob can yield inaccurate answers on the period of one oscillation. This

error can be reduced by letting the pendulum swing for two full oscillations before starting the timer.

Conclusions Overall, the principle or physical law of nature investigated was the scientific method by using the simple pendulum to verify the relationship among the variables and find the value of acceleration due to gravity (g). From the experiment, I learned to apply the scientific method to theoretical predictions to see their correlation, understand how physical parameters were used to investigate theoretical predictions, and understand the approximations to facilitate experimental investigations and analyses.

Questions 1. It was suggested that the time for several periods be measured and the average period determined, rather than timing only one period. a. What are the advantages of this method? The advantage of this method is to increase accuracy and reduce error (lessen systematic error to interfere). b. How and why would the result be affected if a very large number of periods were timed? The angle would change (be inconsistent) as the number of periods increase. Not only would the angle decrease as time goes by, but also the time for one oscillation would increase. Thus, the result would be affected because when amplitude is increased, the velocity increases, which also demonstrates the effects of air resistance.

2. In general, the results of procedure 2 may not have shown clear-cut evidence that the period increases as dramatically with the angle as might be indicated by Eq. 3.1. To understand why, write Eq. 3.1, as

and compute T in terms of T1 for angles of 5, 20, and 60 degrees. Comment on the theoretical predictions and experimental accuracy in relation to your results in Data Table 1.

3. Is air resistance or friction a systematic or a random source of error? Would if cause the period to be larger or smaller than the theoretical value? (Hint: Consider what would happen if the air resistance were much greater, e.g., as though the pendulum were swinging in a liquid.) Air resistance/friction is systematic because it always does the same thing to the length of a period of the pendulum. It always slows the mass, thus making the period longer.

4. Thomas Jefferson once suggested that the period of a simple pendulum be used to define the standard unit of length. What would be the length of a “2-second” pendulum (a pendulum with a period of 2 s)? The period of a pendulum is independent of the mass and the amplitude, if the value of the amplitude is small. It is proportional to the square root of the length.

5. Suppose Jefferson’s 2-second pendulum were operated on the Moon. What would its period be there? It would decrease....