Acceleration due to gravity

Topics: Gravitation, General relativity, Standard gravity Pages: 6 (1167 words) Published: October 19, 2013
Determination of the Acceleration Due to Gravity
By A Good Student
 
Abstract
 
The acceleration due to gravity, g, was determined by dropping a metal bearing and measuring the free-fall time with a pendulum of known period.  The measured value is 9.706 m/s2 with a standard deviation of 0.0317, which does not fall within the range of known terrestrial values.  Centrifugal forces and altitude variations cannot account for the discrepancy.  The calculation is very sensitive to the measured drop time, making it the likely source of error.  

(Short, sweet and to the point.  I give the result, method and comment on its agreement or validity.)  
Theory
 
(First, some background. Be sure to cover any non-numerical aspects of the theory that you wish to address. )  
The acceleration due to gravity is the acceleration experienced by an object in free-fall at the surface of the Earth, assuming air friction can be neglected.  It has the approximate value of 9.80 m/s2, although it varies with altitude and location.  The gravitational acceleration can be obtained from theory by applying Newton’s Law of Universal Gravitation to find the force between the Earth and an object at its surface.  Newton’s Law of Universal Gravitation for the force between two bodies is  

(You may write the equations in by hand.)

where m1 and m2 are the masses of the bodies, r12 is the distance between the centers of mass of the bodies, and G is the Universal Gravitational Constant which has a current accepted value of 6.673 × 10-11 Nm2/kg2.  The force between the Earth and a mass, m, would be where ME and RE are the mass and radius of the Earth, respectively.  For a particular location, G, ME, and RE are constant and may be grouped under a single constant, g.

For obvious reasons, g is sometimes called the local gravitational constant.  It will be numerically equivalent to the acceleration due to gravity on a spherical, non-rotating planet.  (If one evaluates the above using average values from Serway, 6th ed., you obtain g=9.834 m/s2.)  The real acceleration due to gravity will be different than the above due to “centrifugal” and Coriolis effects.  The values that follow were taken from the CRC Handbook of Chemistry and Physics, 75th ed. and illustrate the variability of the value.  As expected, the value is lower at the equator due to centrifugal force.  

(In a real paper, the references would be at the end and would be numbered in the order that they appear in the paper. The citation would simply be the number.  Using in-text citations as I did above will be sufficient for our purposes.)  

(I added this table while writing the Results and Discussion.)  
Location
Value, m/s2
Average value at the equator
9.78036
Average value at the poles
9.83208
Average value over the Terrestrial Ellipsoid
9.7978
 
(The background was a little lengthy in this case.  Now, I start to derive the equations that will be used.  How much of the Theory is spent giving background or deriving will vary.)  
            In this experiment, g was measured using kinematics.  A metal bearing was dropped from a known height and the time was measured.  The kinematic equation that gives position as a function of time is  We will apply this equation to a “drop” (v0 = 0) of height, h, as shown below.  (You may draw diagrams by hand on a separate sheet of paper as long as you refer to them.)  Making these substitutions, we obtain

Rearranging,
(Simple derivation, but still, leave nothing out.  Prove to me that you understand where everything comes from.  I could also add a derivation of the "centrifugal" force to show that it is negligible.) Experimental

 
(Again, you may draw diagrams by hand on a separate sheet of paper as long as you refer to them.) (You may list equipment as a numbered or bulleted list, or in narrative form as done here.  Use past tense.)  

            The solenoid electromagnet was a simple coil of #18 wire with an iron core.  The power...
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