Today ’ s Lab: Measuring Earth Gravity

1. 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!!!

2. 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 27th (Sun.) • Questions? - Hyunju Lee (hyunju@educ.umass.edu) or Stephen Schneider (schneider@astro.umass.edu) Funded by Hubble Space Telescope Education & Public Outreach grant

3. 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

4. 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

5. 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, 0.025 cm or 0.00025 meter • So, for a pendulum 1.5 meter long, we expect the relative error to be:

6. 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 ± εstop) – (t_start ± εstart)

7. 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

8. 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?

9. 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!

10. 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).