Graph made linear:
The period T of a pendulum with length L is given by:
[tex]T=2\pi\sqrt{\frac{L}{g}}[/tex]
To make the equation linear, elevate both members to the 2nd power:
[tex]\begin{gathered} \Rightarrow T^2=(2\pi)^2\cdot\frac{L}{g} \\ \\ \therefore T^2=\frac{(2\pi)^2}{g}L \end{gathered}[/tex]
The variable T^2 depends linearly on the length L. Then, use the values on the table to create a graph of T^2 vs L. To do so, first notice that the table gives values for 10T. Divide the values on the table over 10 to find T, and then square those values to get T^2:
Plot the values of T^2 vs L on a graph:
Slope calculation:
Use a program to find a linear regression for the given data values. This will give us the value of the slope and the y-intercept of the line that best fits the data:
The values of the slope and the y-intercept are:
[tex]\begin{gathered} m=3.17997 \\ b=0.56827 \end{gathered}[/tex]
Comparison of the slope calculation to the expected variable:
For the function:
[tex]T^2=mL+b[/tex]
The value of b tells us that the period of the pendulum would not be equal to 0 when the length of the string is 0, this may be due to the size of the mass hanging from the string.
On the other hand, the slope of the line corresponds to the coefficient of L:
[tex]T^2=\frac{(2\pi)^2}{g}L[/tex]
Then:
[tex]m=\frac{(2\pi)^2}{g}[/tex]
Then, we can calculate the value of the gravitational acceleration using the slope m. As measured by this experiment, the value of g is:
[tex]g=\frac{(2\pi)^2}{m}=\frac{(2\pi)^2}{3.17997\frac{s^2}{m}}\approx12.4\frac{m}{s^2}[/tex]
The expected value of g is 9.81 m/s^2. Then, the percent error between our measurement and the accepted value is:
[tex]\frac{12.4-9.81}{9.81}\times100\%=26.4\%[/tex]
Conclusion:
We can conclude that the value of the gravitational acceleration according to this experiment, is 12.4m/s^2, which exceeds by 26.4% the accepted value of 9.81m/s^2 The order of magnitude of the measurement is correct, so, another conclusion is that a pendulum can be used to measure the gravitational acceleration.
