Download 2010 AP Gravitation Notes

2010 2nd Trimester AP gravitation chapter 13
AP Gravitation
1. What did Kepler discover about the motion of the planets?
First Law.
Second Law
Third Law
2. A satellite is placed in a circular orbit about Earth with a radius equal to one-half
the radius of the Moon’s orbit. What is its period of revolution in lunar months?
(A lunar month is the period of revolution of the Moon.)
3. An asteroid, whose mass is 2.0 x 10-4 times the mass of Earth, revolves in a
circular orbit around the Sun at a distance that is twice Earth’s distance from the
Sun. Calculate the period of revolution of the asteroid in years.
2010 2nd Trimester AP gravitation chapter 13
4. How did Newton expand on Kepler’s Laws? Derive Newton’s Law of
Gravitation.
5. A spherical, nonrotating planet has a radius R and a
uniform density ρ throughout its volume. Suppose a
narrow tunnel were drilled through the planet along
one of its diameters, as shown in the figure above,
in which a small ball of mass m could move freely
under the influence of gravity. Let r be the
distance of the ball from the center of the planet.
a. Find the force due to gravity inside the planet.
b. On the axes below, sketch the force F on the ball as a function of distance r from
the center of the planet.
c. What physics quantity is the area of the above graph?
2010 2nd Trimester AP gravitation chapter 13
d. What if Earth was a shell, what would the force of gravity be inside the shell?
6. If F1 is the magnitude of the force exerted by the Earth on a satellite in orbit about
the Earth and F2 is the magnitude of the force exerted by the satellite on the Earth,
then which of the following is true?
a. F1 is much greater than F2
b. F1 is slightly greater than F2
c. F1 is equal to F2
d. F2 is slightly greater than F1
e. F2 is much greater than F1
7. Example
Alice (50kg) and Arlo(100kg) are 20 meters apart. What is the gravitational force
between them?
8. Find the gravitational force on M2. M1=M3=10 kg, M2= 20 kg
M1
1m
M2
1m
M3
9. A spaceship is on a straight-line path between Earth and its moon. At what
distance from Earth is the net gravitational force on the spaceship zero?
2010 2nd Trimester AP gravitation chapter 13
10. A student is given the set of orbital data for some of the moons of Saturn shown
below and is asked to use the data to determine the mass of Saturn. Assume the
orbits of these moons are circular.
Orbital Period, T
Orbital Radius R,
(seconds)
(meters
8.14 x 104
1.85 x 108
1.18 x 105
2.38 x 108
5
1.63 x 10
2.95 x 108
2.37 x 105
3.77 x 108
a. Use the assumption of circular orbits to derive an equation for the orbital period T
of a moon as a function of its orbital radius R.
b. Which quantities should be graphed to yield a straight line whose slope could be
used to determine Saturn’s mass?
11. A uniform solid sphere of radius R produces a gravitational acceleration of ag on
its surface. At what two distances from the center of the sphere is the
gravitational acceleration ag/3?
2010 2nd Trimester AP gravitation chapter 13
12. A satellite moves in a stable circular orbit with speed vo at a distance R from the
center of a planet. For this satellite to move in a stable circular orbit a distance 2R
from the center of the planet, the speed of the satellite must be
v0
a. 2
v0
b.
c.
2
vo
2v 0
d.
e. 2vo
D
v
M
M
v
13. Two identical stars, a fixed distance D apart, revolve in a circle about their mutual
center of mass as shown above. Each star has mass M and speed v. G is the
universal gravitational constant. Which of the following is a correct relationship
among these quantities?
a. v2=GM/D
b. v2=GM/2D
c. v2=GM/D2
d. v2=MGD
e. v2=2GM2/D
14. Two artificial satellites, 1 and 2, orbit the Earth in circular orbits having radii R1
and R2, respectively, as shown above. If R2 = 2R1, the accelerations a2 and a1 of
the two satellites are related by which of the following?
(A) a2 = 4a1
(B) a2 = 2a1
(C) a2 = a1
(D) a2 = a1/2
(E) a2 = a1/4
2010 2nd Trimester AP gravitation chapter 13
15. The mass of Planet X is one-tenth that of the Earth, and its diameter is one-half
that of the Earth. The acceleration due to gravity at the surface of Planet X is
most nearly
a. 2m/s2
b. 4m/s2
c. 5m/s2
d. 7 m/s2
e. 10 m/s2
16. A newly discovered planet has twice the mass of the Earth, but the acceleration
due to gravity on the new planet’s surface is exactly the same as the acceleration
due to gravity on the Earth’s surface. The radius of the new planet in terms of the
radius R of Earth is
a. ½ R
2
R
b.
2
c.
2R
d. 2R
e. 4R
17. A newly discovered planet, "Cosmo," has a mass that is 4 times the mass of the
Earth. The radius of the Earth is Re. The gravitational field strength at the surface
of Cosmo is equal to that at the surface of the Earth if the radius of Cosmo is
equal to
a. ½Re
b. Re
c. 2Re
d. R e
e. Re2
18. At the surface of the Earth is the only acceleration we have that due to gravity?
Why or why not?
2010 2nd Trimester AP gravitation chapter 13
19. Three uniform spherical planets that are identical in size and mass. The periods
of rotation T for the planets are given, and six lettered points are indicated- three
points are on the equators of the planets and three points are on the north poles.
Rank the points according to the value of the free fall acceleration g at them,
greatest first.
16h
24h
48h
b
d
f
a
c
e
20. Find the acceleration at the equator of the Earth.
21`. Two stars, A and B. are in circular orbits of radii ra and rb, respectively,
about their common center of mass at point P, as shown above. Each star has the
same period of revolution T.
Determine expressions for the following three quantities in terms of ra, rb, T, and
fundamental constants.
a. The centripetal acceleration of star A
2010 2nd Trimester AP gravitation chapter 13
b. The mass Mb of star B
c.
The mass Ma of star A
Determine expressions for the following two quantities in terms of Ma, Mb, ra, rb,
T, and fundamental constants.
d. The moment of inertia of the two-star system about its center of mass.
e.
The angular momentum of the system about the center of mass.
2010 2nd Trimester AP gravitation chapter 13
22. When the satellite is at the desired altitude over the equator, its velocity
vector has the correct direction, but the speed is slightly faster than the
correct speed for a circular orbit of that radius. Draw the orbit below.
ii. When the satellite is at the desired altitude over the equator, its velocity vector
has the correct direction, but the speed is slightly slower than the correct speed
for a circular orbit of that radius. Draw the orbit below.
Force
0
Re
R2
R
23. The graph above shows the force of gravity on a small mass as a function of its
distance R form the center of the Earth of radius Re, if the Earth is assumed to have a
uniform density. The work done by the force of gravity when the small mass
approaches Earth from far away and is placed into a circular orbit of radius R2 is best
represented by the area under the curve between
a. R=0 and R=Re
b. R=0 and R=R2
c. R=Re and R=R2
d. R=Re and R= 
e. R=R2 and R= 
2010 2nd Trimester AP gravitation chapter 13
24. Find the gravitational potential associated with Newton’s Law of Universal
gravitation inside and outside a planet.
25. Determine the speed of the ball when it reaches the
center of the planet.
26. When an object is moved from rest at point A to rest at point B in a
gravitational field, the net work done by the field depends on the mass of
the object and
a. the positions of A and B only
b. the path taken between A and B only
c. both the positions of A and B and the path taken between them
d. the velocity of the object as it moves between A and B
e. the nature of the external force moving the object from A to B
27. A small mass is released from rest at a very great distance from a larger stationary
mass. Which of the following graphs best represents the gravitational potential
energy U of the system of the two masses as a function of time t?
(A)
U
(B)
U
(C)
O
O
t
t
(D)
O
U
O
U
(E)
t
U
O
t
t
2010 2nd Trimester AP gravitation chapter 13
28. Find the speed an object would have to attain to escape the gravitational pull of
an astronomical body of mass M and radius R.
29. The escape speed for a rocket at Earth’s surface is ve. What would be the rocket’s
escape speed from the surface of a planet with twice Earth’s mass and the same
radius as Earth?
a. 2ve
b.
2ve
c. ve
ve
d.
2
e. ve/2
30. Find the kinetic energy associated with a circular orbit.
31. Find the Total Mechanical energy of an orbiting satellite in a circular orbit.
32. What is the total mechanical energy of an orbiting satellite in an elliptical orbit?
33. Why does the gravitational force do work in an elliptical orbit, but not in a
circular orbit? What is the net work done for one complete elliptical orbit?
2010 2nd Trimester AP gravitation chapter 13
34. Rank the positions of a satellite in elliptical orbit from greatest to least.
a. gravitational force
b. speed
D
c. momentum
C
d. Kinetic Energy
e. Potential energy
f. total energy
B
g. acceleration
A
35. A space shuttle is initially in a circular orbit of radius r about Earth. At point P,
the pilot briefly fires a forward-pointing thruster to decrease the shuttle’s kinetic
energy K and mechanical energy E.
2
1
P
Earth
a. Which of the dashed elliptical orbits shown in the figure will the shuttle
then take?
b. Is the orbital period T of the shuttle then greater than, less than, or the
same as in the circular orbit?
36. In March 1999 the Mars Global Surveyor (GS) entered its final orbit about Mars,
sending data back to Earth. Assume a circular orbit with a period of 1.18 x 102
minutes = 7.08 x 103 s and orbital speed of 3.40 x 103 m/s . The mass of the GS is
930 kg and the radius of Mars is 3.43x106m .
(a) Calculate the radius of the GS orbit.
(b) Calculate the mass of Mars.
(c) Calculate the total mechanical energy of the GS in this orbit.
2010 2nd Trimester AP gravitation chapter 13
(d) If the GS was to be placed in a lower circular orbit (closer to the surface of Mars),
would the new orbital period of the GS be greater than or less than the given period?
______Greater than
_______Less than
Justify your answer.
(e) In fact, the orbit the GS entered was slightly elliptical with its closest approach to Mars
at 3.71 x 105 m above the surface and its furthest distance at 4.36 x 105 m above the
surface. If the speed of the GS at closest approach is 3.40 x 103 m/s , calculate the speed
at the furthest point of the orbit.
36. A satellite of mass m is in an elliptical orbit around the Earth, which has mass Me
and radius Re. The orbit varies from closest approach of a at point A to maximum
distance of b from the center of the Earth at point B. At point A, the speed of the satellite
is vo Assume that the gravitational potential energy Ug = 0 when masses are an infinite
distance apart. Express your answers in terms of a, b, m, Me, Re, vo, and G.
a. Write the appropriate definite integral, including limits, that can be evaluated to show
that the potential energy of the satellite when it is a distance r from the center of the
Earth is given by
Ug = -Gmem
r
b.
Determine the total energy of the satellite when it is at A.
2010 2nd Trimester AP gravitation chapter 13
c.
What is the magnitude of the angular momentum of the satellite about the center of
the Earth when it is at A ?
d.
Determine the velocity of the satellite as it passes point B in its orbit.
As the satellite passes point A, a rocket engine on the satellite is fired so that its orbit is changed to a
circular orbit of radius a about the center of the Earth.
e. Determine the speed of the satellite for this circular orbit.
f.
Determine the work done by the rocket engine to effect this change.