Today’s Lab: Measuring Earth Gravity Today we will make a true measurement and estimate its uncertainty We will measure the gravitational acceleration of Earth We will exploit a handy property of the pendulum: As long as the swing angle of the pendulum is not too big, θ<1°, the period of the pendulum only depends on its length and on Earth’s gravity. And on nothing else. Also, as long as the swing angle of the pendulum is not too big, the oscillations are isochronous, i.e. they always take the same amount of time In other words, the period of the pendulum is always the same, even when the pendulum looses energy (the oscillations become smaller and the speed decreases) due to frictions. Nothing to worry about systematics!!!

Survey – extra credits (1.5pt)! Study investigating general patterns of college students’ understanding of astronomical topics There will be 3~4 surveys this semester. Anonymous survey (the accuracy of your responses will not affect your course grade). But, be accurate, please! Your participation is entirely voluntary. SPARK: Assessments > Survey2 The second survey is due: 11:59pm, March 27 th (Sun.) Questions? - Hyunju Lee or Stephen Schneider Funded by Hubble Space Telescope Education & Public Outreach grant

What Decides The Period of The Pendulum? As long as the swing angle is small, i.e. θ≈1° or less, the period of the pendulum T (in sec) is Where L is the length of the pendulum and g is Earth’s gravity (in meter sec -2 ). L (in meter): the length from the fulcrum to the barycenter of the pendulum mass

Earth’s Gravity Solving for “g”, we find: i.e. all is required to get “g” is to measure the length L and period T of a pendulum

About the Errors: measuring length We have to keep track of our Measure Error when measuring the length: the read-out error We will use a ruler Remember how to estimate the read- out error when using the ruler It is the minimum subdivision we can appreciate with confidence In this case, cm or meter So, for a pendulum 1.5 meter long, we expect the relative error to be:

About the Errors: Measuring time Remember that to get the period, we need to measure the time the pendulum takes to complete one full oscillation. Depending on whether the observer acts too soon or too late, the measure gets altered by an unknown amount. This happens twice: when we start the stopwatch and when we stop it. Typical student’s reaction time: 0.2 sec each time. The total uncertainty on the time measure then is 0.3 sec For a period of about 2.46 sec T = (t_ stop ± ε stop ) – (t_ start ± ε start )

The quality of a measure: the relative, or fractional, error Remember that to express the quality of a measure we take the ratio between the error and the measure itself (relative error): So, what sort of relative error do we expect in our measure of g, namely what is

The Error on the Measure of “g” So, if ΔL is the error on L, and ΔT is the error on T, what is the error on g? From the propagation of errors, we calculate the relative error on g: So, once we measure g, we can also get the total error on g Based on the error estimates before, we expect the relative error on g to be about 0.17, or 17%. Not very good. Note that the total error budget is, by far, dominated by the error on T. How can we minimize it?

Reducing the Error on T The total error on a time measure, 0.3 sec, will not change if the measure is long or short. But the relative error will! It will be smaller for longer measures. So, to reduce the relative error on T we want to measure long T’s. But how, if T is fixed by g and by L? Don’t measure the time needed for one oscillation, measure the time needed for 100 or 200 of them! This simple trick of measuring the time of N oscillations makes us reduce the error on T by N times! (just be careful keeping count…) SOOO… let’s do the measure!

IMPORTANT Keep the log of all your measurements, including the calculations Keep the log of the error analysis You WILL NEED ALL THESE to do the problems Now, record your attendance by clicking either A or B (the instructor will have to set the clickers).