(The following is an example of a full lab report of the type that we will do a few times a
semester. It is for an old experiment that we no longer do. It involves concepts from
motion, Newton's laws and circular motion. Except for a few points, it should be
completely understandable by a student who has completed these chapters. I've included
instructor's notes in blue italic type to highlight several important points.)
Determination of the Acceleration Due to Gravity
By A Good Student with ….list group members
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.)
1
Theory
(First, give 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. Microsoft Word also has an equation editor
that you can use.)
F =G
m1m2
r122
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
F =G
M Em
RE2
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.
 GM
F = m 2 E
 RE

 = mg

For obvious reasons, g is 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 a value of
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.)
2
Location
Average value at the equator
Average value at the poles
Average value over the Terrestrial Ellipsoid
Value, m/s2
9.78036
9.83208
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 using a pendulum as described
in the Experimental section. The kinematic equation that gives position as a function of
time is
1
x(t ) = x0 + v0t + at 2
2
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.)
x0=h, t=0
v0=0
a = -g
x(t)=0
Figure 1
Making these substitutions, we obtain
0=h−
Rearranging,
3
1 2
gt
2
g=
2h
t2
(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.)
Pendulum
Solenoid
Bearing
Solenoid
(a)
(b)
Figure 2
(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 source for both solenoids was a standard LabVolt regulated power supply. The
steel bearing had a diameter of 1.6 cm and a mass of 28.4 g. The physical pendulum
consisted of an aluminum rod which is weighted at the bottom. A stopwatch was used to
record the period of the pendulum. The distance was measured with a standard meter
stick.
4
The period of the pendulum was measured by measuring the time for five
oscillations and dividing. The experiment was arranged as shown in Figure 2a. The
pendulum was pulled away from equilibrium and held in place by an electromagnet. The
bearing was held in place by another solenoid wired to the same power supply. A piece
of spark tape was attached to the inside surface of the pendulum. When the power supply
was shut off, the bearing and pendulum were released simultaneously. The bearing
contacts the pendulum as shown in Figure 2b, leaving a mark on the tape. The distance
was then measured. After several calibration runs, ten experimental runs were
performed. The results were obtained utilizing the same time and ten measured
distances.
Data and Analysis
Time for 5 Periods:
Time for 0.25 Periods:
11.50 s
0.575 s
Calculated g, m/s2
9.733
9.745
9.678
9.696
9.702
9.714
9.714
9.751
9.660
9.666
9.706
0.0317
Height, m
1.609
1.611
1.600
1.603
1.604
1.606
1.606
1.612
1.597
1.598
Average
Standard Deviation
(At this point, you would show all calculations.)
Results and Discussion
(I considered putting the literature data from the Handbook of Chemistry and Physics in
the discussion below, but decided to go back and put it in the Theory section.)
The acceleration due to gravity was measured to be 9.706 m/s2 with a standard
deviation of 0.0317. The values quoted in the theory section show that this measurement
is well outside the expected range. The difference between the equator and the poles is
only about 0.05, and these values differ from 9.80 by only 0.02 to 0.03. The values from
the literature account for centrifugal force, but not altitude. A quick calculation would
show that this is also negligible. If we recalculate the value of g from the theory section
by adding 10 km to the Earth’s radius, we obtain a value that differs by only 0.03.
5
GM E
g=
(RE + h )2

Nm 2 
 6.673 × 10 −11
 5.98 × 10 −11 kg
2 
kg 
=
2
6.37 × 106 m + 10000 m
= 9.803 m / s
(
(
2
)
( as compared to 9.834 m/s )
)
2
The calculation of g from our measurements is very sensitive to time since it is
squared in the calculation. We can recalculate g using the distance from run 1 to see how
it might affect the answer. If we vary the time for a quarter oscillation of the pendulum
by just 0.03 s, we obtain
Time, s (at
h=1.609 m)
0.572
0.575
0.578
Calculated g, m/s2
9.84
9.73
9.63
A difference of 0.03 s would correspond to a difference of 0.03 ×4×5=0.6 seconds in the
measurement of five oscillations. While it is difficult to imagine we could be off by over
half a second, the measurement of time clearly deserves more attention in future
experiments. Human reaction time is already a few tenths of a second. Future
experimental designs should seek to measure the time more accurately. More oscillations
would make this measurement more accurate, but damping might become as issue. This
suggests that an electronic method should be used. A good future experiment would be
to measure the altitude variation of g, but this would require better accuracy than the
current experiment.
(Did I leave anything out?)
6