Measuring 'g'
Objectives In this lab, you will measure 'g', the acceleration due to gravity on earth. You will get practice in collecting and interpretting data and estimating uncertainty. You will also use your measurement of g to find the earth's mass. Suggested Reading 5.1 "Describing Motion: Examples from Daily Life", 5.2 "Newton's Laws of Motion" (AKA "Understanding Motion...") 5.3 "The Force of Gravity", and 5.6 "The Acceleration of Gravity"

Materials:

• Golf ball
• Stop watch with 1/100 of a second precision
• String and rulers to measure height

Procedure:

• Find a location suitable for dropping your small object. Drop from a height that is high enough to minimize the relative fraction of time it takes to start and stop the stopwatch but not so high that air resistance starts to affect the result (ie- the object should not achieve terminal velocity). One story high is a good compromise.
• Measure the distance the object will fall - in meters. Everything in this class is in metric units!
• Time the object's fall at least 20 times.
• Calculate the earth's acceleration due to gravity using the relation (derived from calculus)

       g = 2h / t²

  where h is the height of the fall and t is the average time of fall.

• Use your spread in values of t to estimate your random uncertainty:

       Δt = (longest time - shortest time) / 2.

• If there were any sources of error note them and make estimates of how much they might have affected your data. In this lab you might be tempted to blame some of your uncertainty on "human error". Please keep in mind that you need to be able to quantify all sources of uncertainty.
• Use these uncertainties to estimate the biggest and smallest possible values you could get for g:

       g_low = 2h / ( t + Δt )²
       g_hi = 2h / ( t - Δt )²

  This gives you the range of values of g that are allowed within your uncertainty.

Questions:

1. Quantitatively compare your value for g to the current best estimate for the earth's acceleration due to gravity (which you can find in Appendix A of your book). Consider your uncertainty when you compare.
2. Were there any other sources of error in your estimate of g. besides the uncertainty in t?
3. Physicists have discovered that the acceleration due to gravity on a planet's surface is related to the planet's mass and radius:

        g = G M / R²

   where G is the gravitational constant (equal to 6.67x10^-11 N m² / kg²), R is the planet's radius and M is its mass. Using the shadows of poles in two different latitudes, someone else has estimated the radius of the earth to be 6.37x10⁶ m. Calculate the mass of the earth.

4. Use your uncertainty in g to estimate your uncertainty in M.
5. Quantitatively compare this value to the current best estimate for the earth's mass (also in Appendix A of the book). Consider your uncertainty in M. Does your value match if you include the uncertainty?

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Astro101