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# Lab Report: The Simple Pendulum

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Gain insight on how scientists come to understand natural phenomena through theoretical and experimental data by determining the Period of a Simple Pendulum. This experiment will introduce us to the processes of data collection and the procedures used for data /error analysis.

Theory:

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A Period of motion is a physical quantity associated with any cyclical natural phenomenon and is defined as one complete cycle of motion. There are many examples of this in nature, such as the earth’s period of rotation around the sun takes approximately 365 days.

The Simple Pendulum is a basic time-keeping apparatus. A weight is suspended on a length of string which in turn is attached to a frictionless pivot so it can swing freely. The time period it takes to complete one swing is determined by the theoretical equation derived from the Physical Theory of Repeating Motions, aka Simple Harmonic Motion.

T=2π〖[L⁄g]〗^(1/2)

Where T is the period, L is the length of the pendulum and g is the acceleration due to gravity, g=9.

81 m/s^2.

Once finding the theoretical period we when can compare it to experimental measured value we found of the period. In gathering the experimental data there will be a degree of uncertainty associated with the gathered values. Because of the uncertainty in gathering data this must also be applied to the theoretical value of T with the following equation.

∆T=T〖[((1⁄2 ∆L)/L)^(2 )+ ((1⁄2 ∆g)/g)^2]〗^(1/2)

∆T represents the uncertainty of the value T, ∆L represents the uncertainty in the length of string and ∆g represents the uncertainty in the acceleration from gravity.

With these equations we can compare the theoretical value of T with experimental values of T and can find the statistical uncertainly of our results. If the theoretical and experimental values are equal within the uncertainty ranges we can say that the theory has been proven valid. If they aren’t equal there was some error in the experiment or the theory itself.

Equipment and Procedures:

The equipment used in this experiment was a simple pendulum, a wooden meter stick, a PASCO photogate timer, and a digital stopwatch. Our lab assistant had the equipment setup prior to lab so we didn’t assemble it ourselves.

We first used the wooden meter stick to measure the string from where it was tied at the pivot to around the center of the weight and recorded this length, L, in meters. Next we calculated the theoretical period of our simple pendulum. Then proceeded to find experimental values of T with two different methods of measurement: a PASCO photogate timer and a stopwatch.

The PASCO photogate timer records the period of the pendulum directly with the senor at the base of the pendulum and the sensor at precisely the center of where the weight hangs when not swinging. Once the pendulum starts swinging the values are recorded in a table on the computer screen. We recorded this information and calculated the Standard Deviation of the average.

Next we measured the period with a digital stopwatch. One person started the pendulum swinging and another timed 5 oscillations with the stopwatch. We divided the time by 5 and recorded it. This was repeated until we had five eriods at which point we calculated the average of the 10 measurements and calculated the Standard Deviation of the average.

Raw Data:

Measurements and derived values from the PASCO photogate timer iPeriod T_i (s)T_i – (s)〖(T_i-)〗^2 〖(s)〗^2
11.7055.00074.9×〖10〗^(-7)
21.7047-.00011.0×〖10〗^(-8)
31.7054.00063.6×〖10〗^(-7)
41.7042-.00063.6×〖10〗^(-7)
51.7045-.00039.0×〖10〗^(-8)
61.7049.00011.0×〖10〗^(-8)
71.7044-.00041.6×〖10〗^(-7)
81.7043-.00052.5×〖10〗^(-7)
91.7049.00011.0×〖10〗^(-7)
101.7048.00000

Measurements and derived values from the stopwatch
iPeriod T_i (s)T_i – (s)〖(T_i-)〗^2 〖(s)〗^2
11.732.016002.560×〖10〗^(-4)
21.694-.019003.610×〖10〗^(-4)
31.712-.0040001.600×〖10〗^(-5)
41.688-.028007.840×〖10〗^(-4)
51.706-.010001.000×〖10〗^(-4)
61.756.040001.600×〖10〗^(-3)
71.700-.016002.560×〖10〗^(-4)
81.744.0028007.840×〖10〗^(-4)
91.720.0040001.600×〖10〗^(-5)
101.708-.0080006.400×〖10〗^(-5)

Calculations and Error Analysis:

1) Calculating the theoretical period of the simple pendulum.

First use the Simple Harmonic Motion equation,

T=2π〖[L⁄g]〗^(1/2)

L = .725 m (measurement from meter stick)
∆L=±.002 m (given information)
g=9.81 m/s^2 (given information)

T=2π〖[.725m⁄(9.81 m/s^2 )]〗^(1/2)=1.708 s

Then calculate the uncertainty of ∆T with this equation,

∆T=T〖[((1⁄2 ∆L)/L)^(2 )+ ((1⁄2 ∆g)/g)^2]〗^(1/2)

L = .725 m (measurement from meter stick)
∆L=±.002 m (given information)
g=9.81 m/s^2 (given information)
∆g=±.002 m/s^2 (given information)
T=1.708 s (calculated)

∆T=1.708s〖[((1⁄2*.002 m )/(L.725 m))^(2 )+ ((1⁄2*002 m/s^2 )/(9.81 m/s^2 ))^2]〗^(1/2)=2.36 ×〖10〗^(-3) s

Therefore, T= 1.708± 2.36 ×〖10〗^(-3) s

2) Measuring the period of the simple pendulum with PASCO photogate timer.

Calculate the average value of the Period T_i with = (∑_i▒〖=1(T_1 ) 〗)/N , where ∑_i▒〖=1(T_1 )=17.0476〗 and N=10. So = 17.0476/10=1.7048

Calculate the Standard Deviation σ of the average with this equation.

σ= (∑_i▒〖=〖(T_i-)〗^2 〗)/(N-1)

By first finding 〖(T_i-)〗^2 〖(s)〗^2 for each of the ten measurements, then
find the sum of these ∑_i▒〖=〖(T_i-)〗^2 〖(s)〗^2=1.74〗×〖10〗^(-6) 〖(s)〗^2

Next plug in known value’s into the equation to find σ.

σ= (1.74×〖10〗^(-6))/(10-1)=4.397×〖10〗^(-4)

Use σ to estimate the uncertainty in the average period, 〖∆T〗_r.

〖∆T〗_r=σ/√10= (4.397 ×〖10〗^(-4))/√10=1.39×〖10〗^(-4)

Find the systematic uncertainty, 〖∆T〗_s

〖∆T〗_s/t=1hr/1year×(1 year)/(365 days)×(1 day)/(24 hr)

t=1.70476

〖∆T〗_s=1.14155×〖10〗^(-4)*1.70476=1.946×〖10〗^(-4)

Then find the total uncertainty〖∆T〗_tot in the measured period.

〖〖〖∆T〗_tot=[(〖∆T〗_r)〗^2+〖(〖∆T〗_s)〗^2]〗^(1/2)

〖〖〖∆T〗_tot=[(1.39×〖10〗^(-4))〗^2+〖(1.946×〖10〗^(-4))〗^2]〗^(1/2)=2.39×〖10〗^(-4)

Then find, 〖∆T〗_photo
〖∆T〗_photo=±〖 ∆T〗_tot

〖∆T〗_photo=1.70476±2.39×〖10〗^(-4)

3) Measuring the period of the simple pendulum with a stopwatch.

Follow the same steps and equations as described in step 2 to get the total uncertainty 〖∆T〗_tot for the stopwatch.

〖〖〖∆T〗_tot=[(6.862×〖10〗^(-3))〗^2+〖(1.716×〖10〗^(-2))〗^2]〗^(1/2)=1.848×〖10〗^(-2)

And find the 〖∆T〗_stop

〖∆T〗_stop=1.716±1.848×〖10〗^2

Results and Discussion:

Tabulation of the final results of the simple pendulum lab:

T±∆T
Theoretical1.708±.002 s
Photogate1.705±2.39×〖10〗^(-4) s
Stopwatch1.716±1.848×〖10〗^(-2) s

The experiment doesn’t verify the simple harmonic equation, T=2π〖[L⁄g]〗^(1/2) because the experimentally periods do not agree with the theoretically determined period within the limits of uncertainty. The lowest Theoretical time (including -.002 uncertainty) is 1.706, and the closest Experimental time (including uncertainty 2.39×〖10〗^(-4)) is 1.705. The experimental results fall .001s outside the limits of uncertainty.

Of the two experimental methods of gathering data, the photogate has the least overall uncertainty. It was the closest to falling inside the limits of uncertainty with only a .001s difference. The stopwatch was more inaccurate leaving a difference of .004s (including -.002 uncertainty).

The experiment could be improved with more experience in using the equipment (starting the pendulum swing at the same measure of degrees for both the stopwatch and photogate) and/or having more experimental periods recorded. Also there is a degree of probability that there was a mistake made in taking down the recorded results in the first place, it’d be best to have the same lab partner record all the results in an experiment so the data is measured with a more consistent degree of uncertainty.

Conclusion:
The experiment was successful in showing how much human caused uncertainty can affect the final results in an experiment. This was especially apparent with the difference between measuring periods with a stopwatch versus the photogate. There is too much room for error when relying on a human watching the pendulum and punching a stopwatch at the exact same point in the swing versus the photogate automatically and more accurately recording time at a fixed location.

The experiment was not successful in verifying the simple harmonic equation, T=2π〖[L⁄g]〗^(1/2), but was very close in doing so. It may have failed due to one or multiple errors. We could have been a .01 or .001 off when recording periods. I could have made calculator errors during any of the calculations. Since the calculations build on each other one mistake can significantly change the results. Since the results fell outside the limits of uncertainty by a difference of .001 s I would conclude that it was likely a rounding error at some point along the way.

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