The Acceleration Due to Gravity
Prof. M.L.C. Murdock
version: Sept2015
Introduction
To sensibly describe the motion of objects in our universe, we need to understand displacement, velocity
and acceleration. This lab will emphasize the latter. When we use the word “acceleration” we mean the rate
at which the velocity of a moving object changes with time. Accelerations are always caused by “forces.”
(For those who missed lecture, think of a force as a push or a pull.) This is the essence of Newton’s first
law. In today’s lab we will measure the acceleration due to the gravitational force exerted by the earth on
two different types of objects, a tennis ball and a ping-pong ball.
Theory
Newton's second law of force relates the amount of force on an object to its mass and acceleration.
F = ma
(1)
The greater the force on an object, the larger is its acceleration. Beyond that, this equation doesn’t say all
that much until we know what to write on the left-hand side! Fortunately, that is the case for today.
Probably the most apparent (and yet the weakest!) of the known forces in nature is the gravitational force.
Newton's Universal Law of Gravitation describes the mutual attractive gravitational force (Fg) exerted on
each other by two objects with masses m and m’ that are separated by a distance r. The magnitude of this
force is simply
Fg = G m(1) m(2)
(2)
d^2
The constant G is called the gravitational constant and, as its name implies, does not depend on variables
like mass or distance. This course will not explain where this formula comes from or even the experimental
tests done to verify it. Instead, we will
just use it.
To compute the magnitude of the gravitational force between the earth and an object, we substitute the
mass of the earth (ME) and the distance from the object to the center of the earth (r). When the objects are
on or near the earth's surface, this distance can be approximated by the value for the radius of the earth
(RE), so that Equation (2) becomes
Fg =
(3)
We see that the force on the object depends only on the mass of the object, because G, ME, and RE are all
constants and remain the same if we change our object. This force (measured at the earth's surface) is called
the weight of the object.
Looking at Equation (1) and equating F to the gravitational force (Fg), we see that:
ma =
= mg ,
(4)
where
a=
.
This quantity g is a special acceleration and is itself a constant because it depends on quantities that do not
change with time. We call this special acceleration the gravitational acceleration and is, for example, the
same acceleration a book experiences when you drop it.
Allegedly, Galileo first demonstrated this result when he dropped cannonballs of different masses (weights)
from the Leaning Tower of Pisa to show that although they had different masses, when dropped together,
they landed together. This happened in this manner because they both experienced the same acceleration. A
similar experiment can be done by dropping a coin and a feather. When dropped in air, the coin always
lands first, but when they are dropped in a vacuum, an environment where there is no air, they land
together! In the coin and feather case, the different velocities are due to the presence of a force, the
frictional force on the coin and the feather due to the presence of air. Our Equation (4) equates the total
force to the gravitational force and therefore neglects the effects of air friction.
Let’s now try to discover a quantitative method for determining the gravitational acceleration, g. We first
look at the equation for displacement x in a one dimensional universe:
x(t) = vo t + (1/2) at 2
(5)
Here, the quantities vo and a are, respectively, the initial velocity and the acceleration. In this experiment,
like Galileo, we will be dropping an object from rest, so that vo will be zero. We will assume that the only
acceleration present is due to the force of gravity so that we can set a = g. We then obtain the relation that
describes the distance the object falls as a function of time
x(t) = (1/2) gt2 .
(6)
Equation (6) provides a means to measure g. All we need to do is to drop an object through a known
distance x and then measure the time t it takes to fall. If we know both x and t, we can solve equation (6) for
g,
g =
(7)
Hence, by measuring the travel distance and the travel time for an object falling from rest, we can measure
the gravitational acceleration.
Procedure
Data
Use the given lab form to record all data and extended data (t 2 ). You should also record m, the mass of the
ball. Remember to report all data with units. For the last column of all four tables, the value of g to be used
in computing |gave – g| is your value of g found from that particular drop.
Calculations
2
1. Using the measured values for x and the calculated values for t , insert these values into Equation (7) to
get the measured values for g. Show all calculations and remember to label the appropriate quantities with
their respective units.
2. Find the average of the five measurements of g and record this in the table below.
3. As a measure of precision, we will be using the average deviation from the mean for the measured
value of the gravitational acceleration. Reread the lab write-up for Measurement and Measurement Error to
remind yourself on how to do this type of error analysis. Your final result for the gravitational acceleration
should be reported as:
g = gave ±
As a measure of accuracy, use the given accepted value for g (9.8 m/s2) and your measured average value
of g to compute the percentage error. Again, reread the Measurement and Measurement Error write-up to
refresh your memory on how to do this.
Conclusion
1.
How did the gravitational acceleration vary with the different heights? Did it become smaller, larger or
remain roughly constant? Why do you think this is?
2.
What did you discover in this experiment?
Error Analysis
1.
2.
How well do you think you can measure distance in this lab? Explain.
How well do you think you can measure drop time in this lab? Is this a matter of accuracy, precision or
both? Explain.
3.
Examine equation (7). Do you think it is more important to measure accurately the drop time or the fall
distance of the ball to determine g with the greatest possible accuracy? Explain.
4.
Compare your value of g with the accepted value of 9.8 m/s2. Is your value lower or higher than this
value? Why do you think this is? Try to identify the experimental conditions or errors that may cause
this discrepancy.
Ball type: ______________________
trial
1
X
t
t2
g
|gave-g |
2
3
4
Ball mass
Average measured value for g= _______________
Accepted value for g=
Calculations
_______
% error: _______
Conclusions
1.
2.
3.
Error Analysis
1.
2.
3.
4.
Physics 101
DIY : Make and Use a Barometer to Measure Air Pressure
Overview
Air pressure is the result of the weight of tiny particles of air (air molecules)
pushing down on an area. While invisible to the naked eye (i.e. microscopic),
they nevertheless take up space and have weight. For example, take a deep
breath while holding your hand on your ribs and observe what happens. Did
you feel your chest expand? Why did it expand?
Air pressure expands because the air molecules take up space in your lungs,
causing your chest to expand. Furthermore, air can be compressed to fit in a
smaller volume since there's a lot of empty space between the air
molecules. When compressed, air is placed under high pressure.
Meteorologists measure these changes in the air to forecast weather, and the
tool they use is a barometer. The common units of measurement that
barometers use are millibars (mb) or inches of mercury.
DIY Make a Barometer
A. Materials
o
B. Theory
How does this measure air pressure?
C. Procedure
1. Place the completed barometer and scale in a shaded location
free from temperature changes (i.e. not near a window as sunlight
will adversely affect the barometer's results).
2. In your notebook or the table below, record the current date, time,
the weather conditions, and air pressure (i.e. the level where the
end of the straw measures on the scale).
3. Continue checking the barometer twice a day (if possible) each
day over a four days period.
Date
Data Table
Weather
Air
Time
Conditions Pressure
Date
June 4,
2003
June 4,
2003
June 5,
2003
Sample Data Table
Weather
Air
Time
Conditions Pressure
Clear and
9:30 am
4
Sunny
2:30 pm
Cloudy
3
9:30 am
Rainy
1
Barometer Analysis
Answer the following questions in complete sentences.
1. What problem were you trying to solve with your barometer?
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
2. Were there any changes in the weather during the week?
___________________________________________________________________________
___________________________________________________________________________
3. Did the barometer measurement change when the weather changed? How much?
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
4. Did the barometer measurement change without a change in weather? Why do you think that
happened?
___________________________________________________________________________
___________________________________________________________________________
5. How well did your barometer work?
___________________________________________________________________________
___________________________________________________________________________
6. What would you change if you could design the barometer again?
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
7. How do the barometer measurements help us understand the system of weather around us?
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
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