Solutions to Homework 03
Please answer at least 4 of the following 5 questions correctly for full
credit:
Question 1
On an elliptical orbit: Where does a planet move fastest?
Answer:
According to the second of Kepler's laws, a planet moves fastest when it is
closest to the Sun.
Question 2
Things moving in empty space (moon, stars, galaxies, satellites, ...) do not
experience friction the same way things moving on Earth do. Why will they
still not provide examples for the ideal undisturbed motion postulated in
Newton's first law of motion, the law of intertia?
Answer:
Because they are still subjected to a force: Gravity. Gravitational pulls from
galaxies and stars are present everywhere in the Universe, and the genius of
Newton is clearly visible in his against-common-sense assumption that in the
absence of an external force motion (velocity) would not be altered. Not even
in the skies can we see an example of this simple fact. Although once we
understand how gravity works, we understand why even the heavenly bodies do
not show us an example of un-accelerated motion.
Question 3
If two ice-scaters push apart from each other, they excert forces on each
other that are opposite in direction and equal in strength (Newton's third
law). Let us assume they are initially at rest (and push in the same direction
that the blades of their skates are oriented in...).
After they are done pushing and let go of each other: Who will go
faster: the lighter or the heavier of the two? Please tell us why.
Answer:
The answer can be found by combining Newton's second and third laws: The
third law tells us that the force the two skaters exert on each other are
equal in magnitude and opposite in direction, and the second law tells us that
this force is the product of the change of the skater's velocity and the mass
of the skater.
For the product to remain the same if one factor grows, the other factor has to
become smaller proportionally. So if one skater has a smaller mass (is
lighter), that skater's change in velocity (acceleration) has to be
proportionally larger to produce the same magnitude for the force.
As both skaters start out with an equal velocity of zero, and the lighter one
will experience the larger change of velocity, the lighter skater's final
velocity must be larger than the heavier one's.
Question 4
If you use the units pounds for mass, hours for time, and miles for
distance: What units will the constant in Kepler's third law have? Please
make sure you get the right powers of the basic units that are involved; one
of the units I mention above should not be showing up in your answer.
Answer:
Look at page 7 of lecture 6: The units would have to be hours squared over
miles cubed.
Question 5 (the quantitative one...)
Gravitation is what keeps the planets on their orbits around the Sun, and
satellites as well as the Moon on their orbits around the Earth.
By what factor is the acceleration produced by the Earth's gravitational pull
larger at geostationary satellite orbits than at the moon's orbit?
Use 36 km for the radius of the geostationary orbit and 360,000 km for the
radius of the Moon's orbit. (this is just a little bit less than the closest
the Moon's elliptical orbit gets to earth. At the farthest the Moon is a little
more than 400,000 km away from the earth.)
Answer:
First an admission: the geostationary orbit is 36,000 km away, not 36 km...
So if you do the calculation with the correct numbers, the factor is 100
instead of the 100 million I calculate below with my original wrong numbers.
Look at page 6 of lecture 6: Using Newton's second law and equating it with
the gravitational force we can eliminate the mass of the object at the orbit,
and calculate the acceleration from the mass of the Earth. As we are looking
for a ratio, the mass of the Earth and Newton's gravitational constant G will
appear in both the numerator and denominator and cancel out, and all that is
left for the ratio of the accelerations is the ratio of 360,000 squared over
36 squared, which is 100,000,000. - Do not worry: No calculations in the
midterm!
The deadline for submitting solutions to this homework has passed.
Last modified: Fri Feb 1 10:33:02 MST 2008