Download Measure and Calculate Acceleration Due to Gravity

Measure and Calculate
Acceleration Due to Gravity
Topic
The force of gravity can be calculated for each planet.
Introduction
Pretend that you are standing on Mars holding a basketball in one hand
and a Ping-Pong™ ball in the other. If you dropped both balls at the same
time, which would hit the surface first? On Mars, or on any planet, both
objects would reach the surface at the same time because all objects
accelerate at the same rate. The value of this acceleration for any planet
is known as g. Just as the gravity of a planet is uniform, so is the value of
g for that planet.
The value of acceleration for any planet can be calculated with the
following formula:
g=
G + Mplanet
d2
where G is the Universal Gravitational Constant equal to 6.67 x 10–11
Newton-meters2 (N-m2)/kilogram2 (kg2); Mplanet is the mass of the planet;
and d is the distance from the center of the planet to its surface.
In this experiment, you will take measurements and calculate g for Earth,
then use data to calculate the value of g for the other planets.
Time Required
55 minutes
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MEASURE AND CALCULATE ACCELERATION DUE TO GRAVITY
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Materials
✒ access to a location where you can drop a ball about 10
meters (m) (32.8 feet [ft])
✒
✒
✒
✒
✒
✒
✒
ball
meterstick
string
digital stopwatch
calculator
graph paper
science notebook
Safety Note
Please review and follow the safety guidelines.
Procedure
1.
2.
3.
4.
Go with your teacher to a location where you can drop a ball from a
height of about 10 m (32.8 ft).
Work in groups of three. Designate members of the lab team as the
“dropper,” the “timer,” and the “recorder.”
Use the string and meterstick to determine the exact distance from
the dropping point to the ground. Record the distance in your
science notebook.
The “dropper” carries the ball to the drop point. The “timer” turns on
the stopwatch when the ball is released and stops it when the ball
strikes the ground. The “recorder” writes the time (how long it takes
the ball to hit the ground) in a science notebook (see Figure 1).
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MEASURE AND CALCULATE ACCELERATION DUE TO GRAVITY
Figure 1
5.
6.
7.
Repeat step 4 for a total of three trials.
Add all of the times and divide by 3 to find the average time.
Use the following formula to determine the velocity of the ball
during its fall:
velocity =
distance the ball fell
average time
8.
Record the velocity in your science notebook.
Use the following formula to determine the value of acceleration
due to gravity (g) on Earth:
g=
velocity
average time
Record the acceleration in your science notebook.
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MEASURE AND CALCULATE ACCELERATION DUE TO GRAVITY
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Analysis
1.
On Earth, the acceleration of an object (g) is 9.8 meters per second
squared (m/sec2). How did your calculations compare to this known
value? Suggest some sources of error in your calculations.
2. The value of g varies slightly by location because it is dependent on
the distance from the center of the object to the center of Earth.
Copy Data Table 1 in your science notebook. An object on the
Earth’s surface is 6.38 x 106 m from the center of the planet.
Calculate the value of g for an object on Earth’s surface. The mass
of Earth is 5.89 x 1024 kilograms (kg). Repeat the calculation to find
the value at the other distances given on Data Table 1.
Data Table 1
Height above Earth’s
surface (km)
Distance from Earth’s
center (m)
0 (on the surface)
6.38 x 106
2000
8.38 x 106
4,000
1.04 x 107
6,000
1.24 x 107
8,000
1.44 x 107
3.
4.
5.
g (m/s=2)
Graph the information in Data Table 1. Place distance from the
Earth’s center on the x-axis and acceleration on the y-axis.
Analyze the graph, then complete the following sentence by
underlining the correct word in italics:
As the distance from the center of Earth increases, the value of g
(increases, decreases).
The same equation can be used to calculate the value of g on any
planet if the radius and mass of the planet are known. Data Table 2
shows the radius and mass of each planet. Use these values to
calculate g for each planet. Record your calculations in the last
column of Data Table 2.
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MEASURE AND CALCULATE ACCELERATION DUE TO GRAVITY
Data Table 2
Planet
Radius (m)
Mass (kg)
Mercury
2.43 x 106
3.20 x 1026
Venus
6.07 x 106
4.88 x 1024
Mars
3.38 x 106
6.42 x 1023
Jupiter
6.98 x 107
1.90 x 1027
Saturn
5.82 x 107
5.68 x 1026
Uranus
2.35 x 107
8.68 x 1025
Neptune
2.27 x 107
1.03 x 1026
g (m/s=2)
What’s Going On?
The force that holds together all of the parts of the universe
is gravity. Every object in the universe exerts a gravitational
pull on all other objects. The strength of that pull is directly
proportional to the mass of the objects and inversely
proportional to their distance apart.
Although the force of gravity was a topic of interest to the
earliest philosophers, it was not until the 16th century that
scientists began to piece together some true understanding
of that force. Galileo (1564–1642) was the first philosopher
to try to solve the problem scientifically. He believed that
objects fall to the Earth because of gravity, not because of
their weight as other philosophers suggested. Although he
tested his idea, the problem of air resistance and the
difficulty in measuring the speed of falling objects made it
almost impossible to quantify his ideas. Galileo turned to
pendulums and marbles rolling down inclined planes to help
him find a solution. Eventually, he calculated that all objects
falling to Earth accelerate at a constant rate, 32 ft (9.8 m)
per sec. This means that in the first second, a falling object
travels at 32 ft (9.8 m), in the next second at 64 ft (19.6 m)
per sec, and in the third second 96 ft (29.4 m) per sec.
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MEASURE AND CALCULATE ACCELERATION DUE TO GRAVITY
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Sir Isaac Newton (1642–1727), born on the day Galileo died,
continued on his predecessor’s work. Newton produced a
volume of work that explained the behavior of all objects in
the universe. The volume contained his famous three laws of
motion as well as his law of universal gravitation. This last
law can be stated as a formula that calculates the
gravitational attraction between to objects of certain masses
(m1 and m2):
Fgrav =
G · (m1m2)
r2
where Fgrav is the gravitational force; r2 is the square distance
between m1 and m2; and G is a constant.
At the time, Newton did not know the value of G, but he knew
it would be a very small quantity since the force of gravity is
only obvious to an observer between large objects. It was not
until 1798 that the English physicist Henry Cavendish
(1731–1810) determined this value to be 6.75 x 1011
Newton(N) m2/kg2. Today’s astronomers accept the value to
be 6.67259 x 1011 N m2/kg2.
Want to Know More?
See Our Findings.
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OUR FINDINGS
6.7 MEASURE AND CALCULATE ACCELERATION DUE TO GRAVITY
Suggestion for class discussion: Ask students to describe the
force that keeps them from falling off of the Earth. Have them speculate
about the strength of this force on other planets. Come back to this idea
at the end of the experiment to see if they have any new suggestions.
Teacher notes: You might take students to the football stadium or an
indoor stairwell to perform this experiment.
Analysis
1. Answers will vary but might include inaccurate measurements due to
poor quality equipment or error on the part of the students.
2. g = (6.67 x 10-11 N-m2) + (5.98 x 1024 kg)/ 2 (6.38 x 106 m)
g = 2.45 m/s2
Data Table 1
Height above Earth’s
surface (km)
Distance from Earth’s
center (m)
g (m/s2)
0 (on the surface)
6.38 x 106
9.8
2,000
8.38 x 106
5.7
4,000
1.04 x 107
3.7
6,000
1.24 x 107
2.6
8,000
1.44 x 107
1.9
3. Graphs will show a decrease in g as distance increases.
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4. decreases
5.
Data Table 1
Planet
Radius (m)
Mass (kg)
g (m/s2)
Mercury
2.43 x 106
3.20 x 1023
3.61
Venus
6.05 x 106
4.88 x 1024
8.83
Mars
3.39 x 106
6.42 x 1023
3.75
Jupiter
7.14 x 107
1.90 x 1027
26.0
Saturn
6.02 x 107
5.68 x 1026
11.2
Uranus
2.56 x 107
8.68 x 1025
10.5
Neptune
2.48 x 107
1.02 x 1026
13.3
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SAFETY PRECAUTIONS
Review Before Starting Any Experiment
Each experiment includes special safety precautions that are
relevant to that particular project. These do not include all the basic
safety precautions that are necessary whenever you are working on
a scientific experiment. For this reason, it is absolutely necessary
that you read and remain mindful of the General Safety Precautions
that follow. Experimental science can be dangerous, and good
laboratory procedure always includes following basic safety rules.
Things can happen very quickly while you are performing an
experiment. Materials can spill, break, or even catch fire. There will
be no time after the fact to protect yourself. Always prepare for
unexpected dangers by following the basic safety guidelines during
the entire experiment, whether or not something seems dangerous
to you at a given moment.
We have been quite sparing in prescribing safety precautions for the
individual experiments. For one reason, we want you to take very
seriously every safety precaution that is printed in this book. If you
see it written here, you can be sure that it is here because it is
absolutely critical.
Read the safety precautions here and at the beginning of each
experiment before performing each lab activity. It is difficult to
remember a long set of general rules. By rereading these general
precautions every time you set up an experiment, you will be
reminding yourself that lab safety is critically important. In addition,
use your good judgment and pay close attention when performing
potentially dangerous procedures. Just because the book does not
say “Be careful with hot liquids” or “Don’t cut yourself with a knife”
does not mean that you can be careless when boiling water or using
a knife to punch holes in plastic bottles. Notes in the text are
special precautions to which you must pay special attention.
GENERAL SAFETY PRECAUTIONS
Accidents caused by carelessness, haste, insufficient knowledge, or
taking an unnecessary risk can be avoided by practicing safety
procedures and being alert while conducting experiments. Be sure to
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SAFETY PRECAUTIONS
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check the individual experiments in this book for additional safety
regulations and adult supervision requirements. If you will be
working in a lab, do not work alone. When you are working off-site,
keep in groups with a minimum of three students per groups, and
follow school rules and state legal requirements for the number of
supervisors required. Ask an adult supervisor with basic training in
first aid to carry a small first-aid kit. Make sure everyone knows
where this person will be during the experiment.
PREPARING
• Clear all surfaces before beginning experiments.
• Read the instructions before you start.
• Know the hazards of the experiments and anticipate dangers.
PROTECTING YOURSELF
• Follow the directions step by step.
• Do only one experiment at a time.
exits, fire blanket and extinguisher, master gas and
• Locate
electricity shut-offs, eyewash, and first-aid kit.
• Make sure there is adequate ventilation.
• Do not horseplay.
• Keep floor and workspace neat, clean, and dry.
• Clean up spills immediately.
• If glassware breaks, do not clean it up; ask for teacher assistance.
• Tie back long hair.
• Never eat, drink, or smoke in the laboratory or workspace.
not eat or drink any substances tested unless expressly
• Do
permitted to do so by a knowledgeable adult.
USING EQUIPMENT WITH CARE
• Set up apparatus far from the edge of the desk.
• Use knives or other sharp-pointed instruments with care.
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SAFETY PRECAUTIONS
• Pull plugs, not cords, when removing electrical plugs.
• Clean glassware before and after use.
• Check glassware for scratches, cracks, and sharp edges.
• Clean up broken glassware immediately.
• Do not use reflected sunlight to illuminate your microscope.
• Do not touch metal conductors.
alcohol-filled thermometers, not mercury-filled
• Use
thermometers.
USING CHEMICALS
• Never taste or inhale chemicals.
• Label all bottles and apparatus containing chemicals.
• Read labels carefully.
chemical contact with skin and eyes (wear safety glasses,
• Avoid
lab apron, and gloves).
• Do not touch chemical solutions.
• Wash hands before and after using solutions.
• Wipe up spills thoroughly.
HEATING SUBSTANCES
• Wear safety glasses, apron, and gloves when boiling water.
• Keep your face away from test tubes and beakers.
test tubes, beakers, and other glassware made of
• Use
Pyrex™ glass.
• Never leave apparatus unattended.
• Use safety tongs and heat-resistant gloves.
laboratory does not have heat-proof workbenches, put
• IfyouryourBunsen
burner on a heat-proof mat before lighting it.
care when lighting your Bunsen burner; light it with the
• Take
airhole closed, and use a Bunsen burner lighter in preference to
wooden matches.
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SAFETY PRECAUTIONS
4
• Turn off hot plates, Bunsen burners, and gas when you are done.
flammable substances away from flames and other sources
• Keep
of heat.
• Have a fire extinguisher on hand.
FINISHING UP
• Thoroughly clean your work area and any glassware used.
• Wash your hands.
careful not to return chemicals or contaminated reagents to
• Be
the wrong containers.
• Do not dispose of materials in the sink unless instructed to do so.
up all residues and put them in proper containers for
• Clean
disposal.
of all chemicals according to all local, state, and federal
• Dispose
laws.
BE SAFETY CONSCIOUS AT ALL TIMES!
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