# Calculate the magnitude of the acceleration due to gravity including its uncertainty?

I made a free fall time experiment and now I'm trying to make a report.

The experiment consists of releasing a steel ball from various heights hi and measuring corresponding free fall times τi . We used a variant of the Atwood’s machine for the measurement The information about the release and about the impact is transferred to the timer via connectors

the resolution of the measurement is 0.01[s]

Air resistance is disregarded for calculation

then I have to calculate the magnitude of the acceleration due to gravity including its uncertainty

By comparing this relation $h = 1/2 gt^2 + V_0t$ with this relation $h = a_2t^2+a_1t +a_0$. I got $$g =2a_2,\quad a_2 = h/t^2$$ the initial velocity is considered zero

I'm confused here do i have to evaluate a2 for every h and t in my experiment or calculate the average height and time and then evaluate a2 , because I have to find the uncertainty(error) for the gravity

by comparing my gravity result with the accepted gravity , what are the factors that may affect my result?

• Usually in cases like this the best thing you can do is to try to draw a straight line. In this case, if you plot $h$ along the X axis and $t^2$ along the Y axis, you get a straight line with a slope equal to $\frac12 g$. The more points you have, the smaller the uncertainty on your slope will be. Commented May 15, 2016 at 0:45
• h along the x-axis or y-axis , because than I'll get g/2=t^2/h which is not correct, right?
– jack
Commented May 15, 2016 at 1:08
• and if i calculated it this way , how can i find the uncertainty?
– jack
Commented May 15, 2016 at 1:10
• I'm really confused , when i calculate the gravity of each h and time , I got them all around 9.80 to 9.81 but when I calculated it using the slope formula I got 9.86
– jack
Commented May 15, 2016 at 1:20
• Sorry I confused you. The independent variable is $h$ so it should be plotted along X. The dependent variable is $t^2$ so you plot it along Y. But the slope is of course $2/g$ then... If your fitted slope has a different value than the line drawn from the origin to each of your points, it strongly suggests that there is a (systematic?) offset in your measurement. Either a delay in measuring start or stop time, or a bias in the height measurement. Commented May 15, 2016 at 2:06

The measurement you made is time. When you setup you test, you are using another measurement height. With whatever instrument, measurement has error. This will affect test result. In this case, it is $a_2$ you calculated using the test data. I believe each time you repeat a test, the value $a_2$ varies a bit. Different people doing test will give different values. Or different weather temperature can give different values. This is uncertainty.