Otterbein University Department of Physics
Physics Laboratory 1500-11
EXPERIMENT 1500-11
GRAVITY & PLANETARY MOTION
NAME:
INTRODUCTION
In this lab we will explore Newton’s law of universal gravity and some its consequences,
namely planetary motion described by Kepler’s laws.
How does the force of gravity depend on physical parameters? It is clear that it should be
proportional to the masses of the objects involved, but how does it depend on the distance
between the two objects? From geometrical considerations it seems plausible that the
strength of the force should fall off like the distance squared, because the “sphere of
influence” of a force has a surface that grows like an area, i.e. a distance squared. Newton
himself was able to convince himself that this reasoning is met with empirical success by
considering the influence the Earth has on objects in two very different distances: an
object at the surface of the Earth, and the Moon itself. The former is one Earth radius
away from the center of the Earth and experiences an acceleration due to gravity of 9.8
m/s2. The Moon on the other hand, is many tens of Earth radii away and experiences
centripetal acceleration due to its approximate uniform circular motion around the Earth
in a month. We can retrace Newton’s calculation by noting that the distance to the Moon
was known already to the ancient Greeks. Aristarchus around 270 BC derived the Moon's
distance from the duration of a lunar eclipse. It was observed that during a lunar eclipse,
when the (full) Moon moves through the shadow of the Earth cast by the Sun, the it takes
about three hours to cross the shadow. Assuming that the Moon orbits in a large circle of
radius R around Earth, and that the size of the shadow cast by the Earth is the diameter of
the Earth, or D = 2 rE, we can draw up a relation between the known orbital period T (a
month) and the time t it takes to cross the shadow (3 hours) since the Moon’s speed is
approximately constant, v = 2πR/T, namely 2π R / 2 rE = T/t. From this Aristarchus
obtained: R ~ 60 rE, in reasonable agreement with the modern value.
EXPLORING NEWTON’S LAW OF UNIVERSAL GRAVITY
1. By comparing the acceleration at 1 rE (known!) and at 60 rE (compute the centripetal
acceleration) show that the acceleration, and therefore the force, scales like the inverse
distance squared.
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Otterbein University Department of Physics
Physics Laboratory 1500-11
In its general vector form the law of universal gravity between two objects reads
𝐹⃗12 = −𝐺
𝑚1 𝑚2
2
𝑟12
𝑟⃗21 ,
where r12 is the distance between the objects, and the vector 𝑟⃗21 points from object 2 to
object 1, i.e. the force 𝐹⃗12 of object 2 on object 1 is pointing from object 1 to object 2.
2. Where does the gravitational force the Earth exerts on the Moon point to?
3. Use the law of universal gravity to fill in the following table. The table lists by what
factor the magnitude of the gravitational force between the Sun and the Earth, and their
accelerations due to the gravitational force change if a relevant parameter changes. If no
change occurs, put in “1”.
Change
Distance
doubles
Distance halves
FEarth,Sun
FSun,Earth
Radius of Earth
doubles
Radius of Sun
halves
Sun turns into
Black hole
Mass of Earth
doubles
Mass of Sun
doubles
2
aEarth
aSun
Otterbein University Department of Physics
Physics Laboratory 1500-11
ADDING FORCES VECTORIALLY
The law of universal gravity is a vector law, and as such, it falls under the superposition
principle. This means that in order to calculate the combined influence of two or more
forces on a body, we can (and have to) add the forces vectorially. Consider the following
configuration of two satellites in orbit around the Earth. The satellites both have a mass
of 5000 kg. Assume that at one instant in time, one is directly over the North pole, 2rE
from the center of the Earth, and the other one is over the equator 3rE away. Assume the
position of the second satellite defines the x direction.
4. Determine the total force on the Earth (magnitude and direction) due to the two
satellites.
ORBITS UNDER THE INFLUENCE (OF GRAVITY)
Basically, the law of universal gravity tells us that an object, say, a planet, is attracted to
another object, say, the Sun, by a force that gets the stronger the closer the planet gets to
the Sun, and that the force always points directly towards the Sun. If a planet therefore
would be at rest relative to the Sun, it would start falling towards the Sun, and eventually
fall into the Sun. If there is, however, an initial velocity perpendicular to the direction
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Otterbein University Department of Physics
Physics Laboratory 1500-11
towards the Sun, a stable orbit around the Sun results. The orbits of the planets around
the Sun are ellipses, with shapes and the specific motion of the planet described by
Kepler’s three laws of planetary motion. Although Kepler’s laws are a consequence of
Newton’s law of universal gravity, historically Kepler found his laws first empirically,
and Newton took them as input, helping him to deduce his law of gravity. For the
mathematical description of an ellipse, see figure below.
Consider the simulation of Kepler’s laws by going to the following URL on the
computer: http://www.liceoalberti.it/Galileo/Keplero/kepler.laws.html. Follow the
instructions to put a planet a distance of about 50 A.U. away from the Sun onto the
screen. There are nine physical quantities listed at the side: the semi-major axis (a), the
semi-minor axis (b), the eccentricity (e), the angular momentum (L), the kinetic, potential
and total energies, and the velocities in x and y directions.
5. Planets move in space, i.e. in three dimensions. Why are the orbits displayed using
only two coordinates?
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Otterbein University Department of Physics
Physics Laboratory 1500-11
6. Once you have positioned the blue dot, click on the white initial velocity vector, and
explore what shapes and sizes of orbits you can dial in by choosing magnitude and
direction of the initial velocity. Which of the nine quantities can you change this way?
7. Describe how the orbit changes as you …

...increase the initial velocity

…point it perpendicular to the direction towards the sun

…point it away from the sun

…point it towards the sun
8. So far we have been exploring the shape of the orbit. Now, let’s take a look at the
motion of the planet along the orbit. This is the subject of Kepler’s second law. Create an
orbit with a=50, e =0.5 (approx.). Hit the start button and observe how the velocity of the
planet changes as it moves around its orbit. Is this in accordance with Kepler’s second
law? Explain!
9. The third law took Kepler ten more years of hard work. It relates the size of the orbit to
the period of the planets motion around the Sun. Calculate the period of your planet
around the Sun, assuming that the semi-major axis is given in Astronomical Units (A.U.)
and the period is in Earth years. Note that the distance Earth-Sun is defined to be exactly
one A.U., and that therefore aE3 / PE2 = 1.
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Otterbein University Department of Physics
Physics Laboratory 1500-11
10. Redo the calculation for a planet that is half the distance from the Sun. Does it take
this planet half as long to orbit the Sun? Explain why it is reasonable in the light of
Newton’s law of universal gravitation.
11. According to Kepler III, how does the eccentricity of an orbit influence the orbital
period of a planet?
CONSTRUCTING ELLIPTICAL ORBITS
Let’s review the parameters of an elliptical orbit, see figure above.
12. It is easy to draw an ellipse. Take a piece of butcher paper, and draw an ellipse, i.e.
the orbit of a planet around the Sun, with the help of two push pins and a rope. Aim for a
semi-major axis that is about half the width of the paper. Draw in the position of the Sun.
Attach the drawing to your report.
13. Label your graph, and measure or calculate the semi-major axis and the eccentricity.
14. Label the points where the planet is fastest and slowest.
15. Use the same rope to draw a more eccentric orbit, making sure that the Sun is at the
same point as in the last plot! We will refer to this as the “comet orbit.” Is the semi-major
axis the same? Why or why not?
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Otterbein University Department of Physics
Physics Laboratory 1500-11
POTENTIAL GRAVITATIONAL ENERGY
16. For definiteness, we will work with the following masses: Sun: 2•1030kg, planet
6•1024kg, comet 3•1018kg; radii 700,000km (Sun), 6,000km (planet), 200km (comet).
Choose a scale (e.g. 1 million km per inch) and calculate the potential energies of the
planet-sun and the comet-sun system at one of the two intersection points of the orbits.
What is the relation between the two potential energies?
17. Which of the two systems has the larger total energy E. Why?
18. Describe what we mean by “escape velocity”.
19. Arguing with the conservation of mechanical energy, calculate the escape velocity of
the planet.
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