Roy has a mass of 60 kg. Laurie has a mass of 65 kg. They
are 1.5 m apart.
(a) What is the magnitude of the gravitational force of the earth on
(b) What is the magnitude of Roy's gravitational force on the earth?
(c) What is the magnitude of the gravitational force between Roy
(d) What is the magnitude of the gravitational force between Laurie
and the sun?
During a solar eclipse, the moon, earth and sun all lie on
the same line, with the moon between the earth and sun. Define
your coordinates so that the earth and moon lie at greater x values
than the sun. For each force, give the correct sign as well as the
magnitude. (a) What force is exerted on the moon by the sun? (b)
On the moon by the earth? (c) On the earth by the sun? (d) What
total force is exerted on the sun? (e) On the moon? (f) On the
Suppose that on a certain day there is a crescent moon, and
you can tell by the shape of the crescent that the earth, sun and
moon form a triangle with a 135 °interior angle at the moon's corner.
What is the magnitude of the total gravitational force of the earth
and the sun on the moon?
How high above the Earth's surface must a rocket be in order
to have 1/100 the weight it would have at the surface? Express your
answer in units of the radius of the Earth.
he star Lalande 21185 was found in 1996 to have two planets
in roughly circular orbits, with periods of 6 and 30 years. What is
the ratio of the two planets' orbital radii?
In a Star Trek episode, the Enterprise is in a circular orbit
around a planet when something happens to the engines. Spock
then tells Kirk that the ship will spiral into the planet's surface
unless they can fix the engines. Is this scientifically correct? Why?
(a) Suppose a rotating spherical body such as a planet has
a radius r and a uniform density ρ, and the time required for one
rotation is T. At the surface of the planet, the apparent accelera-
tion of a falling object is reduced by acceleration of the ground out
from under it. Derive an equation for the apparent acceleration of
gravity, g, at the equator in terms of r, ρ, T, and G.
(b) Applying your equation from a, by what fraction is your apparent
weight reduced at the equator compared to the poles, due to the
(c) Using your equation from a, derive an equation giving the value
of T for which the apparent acceleration of gravity becomes zero,
i.e. objects can spontaneously drift off the surface of the planet.
Show that T only depends on ρ, and not on r.
(d) Applying your equation from c, how long would a day have to
be in order to reduce the apparent weight of objects at the equator
of the Earth to zero? [Answer: 1.4 hours]
(e) Observational astronomers have recently found objects they called
pulsars, which emit bursts of radiation at regular intervals of less
than a second. If a pulsar is to be interpreted as a rotating sphere
beaming out a natural "searchlight" that sweeps past the earth with
each rotation, use your equation from c to show that its density
would have to be much greater than that of ordinary matter.
(f) Theoretical astronomers predicted decades ago that certain stars
that used up their sources of energy could collapse, forming a ball
of neutrons with the fantastic density of sim1017 kg/m3. If this is
what pulsars really are, use your equation from c to explain why no
pulsar has ever been observed that flashes with a period of less than
1 ms or so.
You are considering going on a space voyage to Mars, in which
your route would be half an ellipse, tangent to the Earth's orbit at
one end and tangent to Mars' orbit at the other. Your spacecraft's
engines will only be used at the beginning and end, not during the
voyage. How long would the outward leg of your trip last? (Assume
the orbits of Earth and Mars are circular.)
(a) If the earth was of uniform density, would your weight be
increased or decreased at the bottom of a mine shaft? Explain.
(b) In real life, objects weigh slightly more at the bottom of a mine
shaft. What does that allow us to infer about the Earth?
Ceres, the largest asteroid in our solar system, is a spherical
body with a mass 6000 times less than the earth's, and a radius
which is 13 times smaller. If an astronaut who weighs 400 N on
earth is visiting the surface of Ceres, what is her weight?
Solution, p. 284
Prove, based on Newton's laws of motion and Newton's law
of gravity, that all falling objects have the same acceleration if they
are dropped at the same location on the earth and if other forces
such as friction are unimportant. Do not just say, "g = 9.8 m/s2 -
it's constant." You are supposed to be proving that g should be the
same number for all objects.
Solution, p. 284
The figure shows an image from the Galileo space probe taken
during its August 1993 flyby of the asteroid Ida. Astronomers were
surprised when Galileo detected a smaller object orbiting Ida. This
smaller object, the only known satellite of an asteroid in our solar
system, was christened Dactyl, after the mythical creatures who
lived on Mount Ida, and who protected the infant Zeus. For scale,
Ida is about the size and shape of Orange County, and Dactyl the
size of a college campus. Galileo was unfortunately unable to measure
the time, T, required for Dactyl to orbit Ida. If it had, astronomers
would have been able to make the first accurate determination
of the mass and density of an asteroid. Find an equation for
the density, ρ, of Ida in terms of Ida's known volume, V , the known
radius, r, of Dactyl's orbit, and the lamentably unknown variable
T. (This is the same technique that was used successfully for determining
the masses and densities of the planets that have moons.)
Solution, p. 284
If a bullet is shot straight up at a high enough velocity, it will
never return to the earth. This is known as the escape velocity. We
will discuss escape velocity using the concept of energy in the next
book of the series, but it can also be gotten at using straightforward
calculus. In this problem, you will analyze the motion of an object
of mass m whose initial velocity is exactly equal to escape velocity.
We assume that it is starting from the surface of a spherically
symmetric planet of mass Mand radius b. The trick is to guess at
the general form of the solution, and then determine the solution in
more detail. Assume (as is true) that the solution is of the form r=
kt p, where r is the object's distance from the center of the planet
at time t, and k and p are constants.
(a) Find the acceleration, and use Newton's second law and Newton's
law of gravity to determine k and p. You should find that the
result is independent of m.
(b) What happens to the velocity as t approaches infinity?
(c) Determine escape velocity from the Earth's surface.
Astronomers have recently observed stars orbiting at very high
speeds around an unknown object near the center of our galaxy.
For stars orbiting at distances of about 1014 m from the object,
the orbital velocities are about 106 m/s. Assuming the orbits are
circular, estimate the mass of the object, in units of the mass of
the sun, 2 × 1030 kg. If the object was a tightly packed cluster of
normal stars, it should be a very bright source of light. Since no
visible light is detected coming from it, it is instead believed to be
a supermassive black hole.
Astronomers have detected a solar system consisting of three
planets orbiting the star Upsilon Andromedae. The planets have
been named b, c, and d. Planet b's average distance from the star
is 0.059 A.U., and planet c's average distance is 0.83 A.U., where an
astronomical unit or A.U. is defined as the distance from the Earth
to the sun. For technical reasons, it is possible to determine the
ratios of the planets' masses, but their masses cannot presently be
determined in absolute units. Planet c's mass is 3.0 times that of
planet b. Compare the star's average gravitational force on planet
c with its average force on planet b. [Based on a problem by Arnold
Solution, p. 284
Some communications satellites are in orbits called geosynchronous:
the satellite takes one day to orbit the earth from west
to east, so that as the earth spins, the satellite remains above the
same point on the equator. What is such a satellite's altitude above
the surface of the earth?
Solution, p. 284
As is discussed in more detail in section 5.1 of book 2, tidal
interactions with the earth are causing the moon's orbit to grow
gradually larger. Laser beams bounced off of a mirror left on the
moon by astronauts have allowed a measurement of the moon's rate
of recession, which is about 1 cm per year. This means that the
gravitational force acting between earth and moon is decreasing. By
what fraction does the force decrease with each 27-day orbit? [Hint:
If you try to calculate the two forces and subtract, your calculator
will probably give a result of zero due to rounding. Instead, reason
about the fractional amount by which the quantity 1/r2 will change.
As a warm-up, you may wish to observe the percentage change in
1/r2 that results from changing r from 1 to 1.01. Based on a problem
by Arnold Arons.]
Solution, p. 285
Suppose that we inhabited a universe in which, instead of
Newton's law of gravity, we had
where k is some
constant with different units than G. (The force is still attractive.)
However, we assume that a = F/m and the rest of Newtonian
physics remains true, and we use a = F/m to define our mass scale,
so that, e.g., a mass of 2 kg is one which exhibits half the acceleration
when the same force is applied to it as to a 1 kg mass.
(a) Is this new law of gravity consistent with Newton's third law?
(b) Suppose you lived in such a universe, and you dropped two unequal
masses side by side. What would happen?
(c) Numerically, suppose a 1.0-kg object falls with an acceleration
of 10 m/s2. What would be the acceleration of a rain drop with a
mass of 0.1 g? Would you want to go out in the rain?
(d) If a falling object broke into two unequal pieces while it fell,
what would happen?
(e) Invent a law of gravity that results in behavior that is the opposite
of what you found in part b. [Based on a problem by Arnold
(a) A certain vile alien gangster lives on the surface of an
asteroid, where his weight is 0.20 N. He decides he needs to lose
weight without reducing his consumption of princesses, so he's going
to move to a different asteroid where his weight will be 0.10 N. The
real estate agent's database has asteroids listed by mass, however,
not by surface gravity. Assuming that all asteroids are spherical
and have the same density, how should the mass of his new asteroid
compare with that of his old one?
(b) Jupiter's mass is 318 times the Earth's, and its gravity is about
twice Earth's. Is this consistent with the results of part a? If not,
how do you explain the discrepancy?
Solution, p. 285
Where would an object have to be located so that it would
experience zero total gravitational force from the earth and moon?
The planet Uranus has a mass of 8.68 × 1025 kg and a radius
of 2.56 × 104 km. The figure shows the relative sizes of Uranus and
(a) Compute the ratio gU/gE, where gU is the strength of the gravitational
field at the surface of Uranus and gE is the corresponding
quantity at the surface of the Earth.
(b) What is surprising about this result? How do you explain it?