Download The Role of Gravity in Orbital Motion

 The Role of Gravity in Orbital Motion
Part of: Inquiry Science with Dartmouth
Developed by: Christopher Carroll,
Department of Physics & Astronomy, Dartmouth College
Adapted from: “How Gravity Affects Orbits” (Ohio State Univ.)
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Overview
Gravity is the natural phenomenon by which all objects in the universe are attracted to
one another. Gravity allows stars to form from clouds of hydrogen gas, planets to form
from molecules of cosmic dust, and is responsible for the orbits of all celestial bodies.
But, what measurable quantities affect the strength of gravity? In this module, students
explore how gravity affects celestial bodies and their orbits.
Science Standards (NGSS)
MS-ESS1-2 Develop and use a model to describe the role of gravity in the motions
within galaxies and the solar system.
MS-PS2-2 Plan an investigation to provide evidence that the change in an object’s
motion depends on the sum of the forces on the object and the mass of the object.
MS-PS2-4 Construct and present arguments using evidence to support the claim that
gravitational interactions are attractive and depend on the masses of interacting objects.
MS-PS2-5 Conduct an investigation and evaluate the experimental design to provide
evidence that fields exist between objects exerting forces on each other even though
the objects are not in contact.
Focus Question
How does the mass of objects and their distance from each other affect the strength of
gravitational attraction?
Objectives
Through this lesson, students will:
• Construct and test a hypothesis as a team
• Determine the dependence of mass and separation on gravitational strength
• Learn how these same properties affect escape velocity
• Understand gravitational attraction as a field (distorted spacetime)
Background
Gravity acts as an attractive force that operates on all objects with mass. The strength
of the gravitational attraction is dependent on only two variables: the mass of the
objects and their distance of separation. As gravity is an attractive force between all
massive objects, the gravitational field permeates all of space, affecting objects both on
and off Earth. This is how Newton came to understand that gravity was as responsible
for the apple falling from the tree as the Moon orbiting the Earth.
Materials
Fishing line (3 lb. test)
2 wooden batons to hold fishing line (optional: +2 carabiners)
Meter stick
Masking tape
Preparation
Trajectories: With the masking tape, mark an area for the Sun as shown in the diagram
below. Then make three small additional markings following the red line at one half,
one, and two meters
away from the Sun.
Gravity (batons): Drill holes in the batons for the fishing line to pass through. Another
option is to have a carabiner attached to the baton—this makes it easier to switch out
the fishing line during the activity.
Gravity (string): Cut 7x(2-meter) pieces, 1x(4-meter) piece, and 1x(8-meter) piece of
fishing line. Tie each end of fishing line into a loop big enough to fit through the baton/
carabiner. It helps to cut the fishing line a little longer than the distance required to
account for making the loops.
Make one copy of the worksheet for each student and distribute.
Procedure
Warm up: Example of Newton’s First Law of Motion
Using the baton attached to a string, twirl the baton around in a circle. Ask the students
what would happen if you let go of the string. Ask the students what keeps the baton
circling around your hand (A: the tension in the string produces a force). Now imagine
that the string was invisible and that this is the same concept as gravity. This is a
demonstration of Newton’s First Law of Motion: An object at rest stays at rest and an
object in motion stays in motion with the same speed and in the same direction unless
acted upon by an unbalanced force. Review the activity with the students and have
them make predictions based on their intuition and fill in the first table.
Activity: Gravity and Orbits
In this activity, students will assume the roles of the Sun, gravity, and a nearby planet.
Split the class up into groups of 3-5 students. Choose one student to be the Sun,
gravity, and the planet. In the case of 4-5 student groups, additional students can act as
“extra mass” for the Sun in the first few trials. The student representing the Sun should
stand on the “X”. The student representing the planet should begin some distance away
as indicated by the arrows. The student representing gravity should stand along the red
line on the opposite side of the planet’s trajectory, facing the Sun. Both the Sun and
Gravity will hold one baton, both ends attached to the length of string.
The Planet will pass along several trajectories across the path of the Sun at a distance
of (one-half, one, and two meters). At the point of closest approach (reaching the red
line) the planet will experience the force of gravity from the Sun—represented by the
fishing line. When the Planet reaches this point they will grab a baton from Gravity and
continue on their path.
Once under the influence of gravity, the Planet will experience one of three possible
outcomes:
1. The string holds. The force of gravity is strong enough to capture the Planet in
orbit around the Sun.
2. The string breaks. The force of gravity is not strong enough to capture the
Planet, but the straight-line trajectory is changed.
3. The string breaks. The force of gravity is too weak to capture Planet or change
it’s trajectory.
Assessment
Construct Newton’s Law of Universal Gravitation and discuss the affects that a change
in mass and a change in separation distance has on the strength of the field. Calculate
the strength of the gravitational force of the other planets in the Solar System relative to
the Earth (mass: Earth mass, distance: AU).
The Role of Gravity in Orbital Motion
Introduction: Today you will investigate the role gravity plays in orbital motion, like the
Moon and Earth, or Earth and the Sun. The force of gravity acts between these celestial
bodies and changes their motions, depicting Newton’s First Law of Motion.
What is Newton’s First Law of Motion?
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Prediction:
Trial
Sun Mass
Distance
# of string
1
1
2m
1
2
2
2m
2
3
3
2m
3
4
1
2m
1
5
1
1m
1 (4 m fold x4)
5
1
0.5 m
1 (8 m fold x16)
String break?
Path changed?
String break?
Path changed?
Experiment: Planet speed _____________
Trial
Sun Mass
Distance
# of string
1
1
2m
1
2
2
2m
2
3
3
2m
3
4
1
2m
1
5
1
1m
1 (4 m fold x4)
5
1
0.5 m
1 (8 m fold x16)
Follow-up Questions
Compare your initial predictions with the results of your experiment. Does the
dependence on mass and separation distance agree with your predictions?
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Is gravity directly or indirectly proportional to the mass of an object? How do the
results of your experiment support this?
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Knowing how mass relates to the strength of gravity, what would you expect to
find if you increase the mass of the planet instead of the Sun? Would gravity’s
dependence on mass change?
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______________________________________________________________________
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Is gravity directly or indirectly proportional to the distance of separation between
two objects? How do the results of your experiment support this?
______________________________________________________________________
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According to the results of your experiment, can you predict the dependance of
gravity on Mass M and separation distance r? (Hint: For r, look closely at # of
strings)
Fgrav
0000
/
0000
Extra Credit
Add your own extra trials to your Results chart and test what happens with the
dependence on the velocity of the Planet. Keep the mass of the Sun and the
distance of the Planet approach constant and vary only the speed at which the
Planet follows the initial trajectory.
Describe your predictions here:
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How does changing the planet’s velocity affect the results?
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What does this tell you about the role kinetic energy plays in determining the
orbit?
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Extensions
For an object like a planet to escape the gravitational attraction of another object, say
the Sun, the planet must be moving with enough speed or the attractive force will be too
great and the planet will be capture into orbit. The speed needed to escape is called the
“escape velocity.” If the planet gets too close and is not moving fast enough, it will be
capture into orbit around the more massive Sun. If the planet is moving fast enough
then it can escape!
We can derive the escape velocity from conservation of energy. For a “bound
system” (think objects in orbit), the kinetic energy must always be less than the potential
energy. We define kinetic and potential energy as the following:
KE =
1
mv 2
2
PE =
GM m
r
Conservation of energy states that:
(KE + P E)initial = (KE + P E)final
If an object escaped and traveled in infinitely large distance away, then the final kinetic
and potential energy would be zero. In that case the above equation becomes:
(KE + P E)initial = 0
1
GM m
mv 2 =
2
r
r
2GM
vescape =
r
Here we see that given any mass of a larger object M at a given distance r, we can
calculate the speed needed by the smaller object to break out of orbit.
Supplemental escape velocity:
Variable speed: The students can discover their “escape” speed from the
system. Using the 1 meter distance trajectory, have the students rerun the
experiment using different Planet speeds and determine how fast they must
move in order to break the string. The students should find that at slower
speeds, the string does not break and they are captured into orbit, but as they
increase the speed, eventually the string will break and they have reached the
escape velocity for the system.