Finding Pi by using the Period of a Pendulum
OverviewThe equation for the period of a simple pendulum contains Pi. Therefore, by working backwards and measuring all other variables in an experiment, and approximation of the value of Pi can be found. The typical physics equation is accurate within 1% when the pendulum is swung at an angle less than 15 degrees (due to the small angle approximation of sine). However, these calculations need to be more accurate and will need to include the initial angle of each swing. Additionally, because 10 periods will be timed for greater accuracy and friction will slow down the pendulum each swing, the initial angle of the pendulum will slightly decrease each swing it makes. Air resistance also needs to be accounted for.
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Diagram of Pendulum Setup
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The Math
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Step 1: Rearranging the equation for the period of a pendulum, an expression for Pi is found. K is a dimensionless quantity that will be calculated in order to account for the starting angle the pendulum is released and the drag force that will decrease the angle as 10 periods are timed.
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L = length of the pendulum | g = acceleration due to gravity | T = period of 1 swing
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Step 2: In order to account for the initial angle that the pendulum is released from, a power series must be used. This is derived by integrating the angular velocity of the bob, combined with the "Legendre Polynomial" to form a MacLaurin series. To the right is what will be substituted for K in equation 2. Below is the beginning terms of the power series. Note: for calculations to estimate Pi, only the first four terms are used (n=0 to n=3) in the below MacLaruin Series.
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Eq. 3: coefficient K to account for angle
θ = angle of pendulum at any given moment
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Eq. 4: full equation for period, accounting for initial angle (not used in calculations, for example purposes) - uses a maclaurin series for sine
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Step 3: The decrease in the initial angle over time (equation 5) is dependent on the constant C, a number based on the pendulum bob's drag coefficient, area, and mass (equation 6). For a full proof of these equations, see here. Note: because the initial angle should be the average initial angle of all ten timed periods, half of the time that the pendulum swings for all 10 periods will be used in equation 5, so the 2 will be changed to a 4. |
θo = initial angle of pendulum's first swing
t = time of 10T | ρ = air density | m = mass of bob A = cross sectional area of bob | Cd = coefficient of drag |
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Step 4: Putting it all together (substituting in the expressions), the comprehensive equation is found (Equation 7). Note that only the first four terms of the MacClaurin series are used in the Pi estimation. Also note that the procedure breaks this equation into sections (in Excel) in order to approximate Pi. |
Eq. 7: entire expression for pi
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Procedure
diagram of setup
hid piest so no influence on timing/procedure/human error
waited to start timing after one swing
Predetermined number of times:
hid piest so no influence on timing/procedure/human error
waited to start timing after one swing
Predetermined number of times:
- Small setup: 1 run with a length on every 0.05m interval between 0.05m and 1m
- Medium setup: 1 run with a length on every 0.5m interval between 1m and 4m
- Large setup: 1 run with a length on every 1m interval between 5m and 10m
Data
screenshots of data - screen shot, batch lr crop
View Data as Online Spreadsheet
Download Data as Excel Spreadsheet
Note: Separate trials are listed as separate sheets in spreadsheet
Download Data as Excel Spreadsheet
Note: Separate trials are listed as separate sheets in spreadsheet
Sources of Error
- Incorrect timing of trials due to random variation
- stats chance of occurrence with number of trials
- length random variation
