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 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

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, 0.025 cm or 0.00025 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 ± εstop) – (t_start ± εstart)

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).