# Lab Report: The Simple Pendulum

**Topics:**Measurement, Pendulum, Simple harmonic motion

**Pages:**7 (1538 words)

**Published:**December 14, 2013

Name: XXXXXXX XXXXX XXXXXXX

Date: January 18, 2013

Objective:

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:

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 Periods 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

11.732.01600...

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