Download AP Physics Free Response Practice – Gravitation *1977M3. Two

AP Physics Free Response Practice – Gravitation
*1977M3. Two stars each of mass M form a binary star system such that both stars move in the same circular orbit
of radius R. The universal gravitational constant is G.
a. Use Newton's laws of motion and gravitation to find an expression for the speed v of either star in terms of R,
G, and M.
b. Express the total energy E of the binary star system in terms of R, G, and M.
c.
Suppose instead, one of the stars had a mass 2M.
On the following diagram, show circular orbits for this star system.
2M
d.
Find the ratio of the speeds, v2M/vM.
1984M2. Two satellites, of masses m and 3m, respectively, are in the same circular orbit about the Earth's center, as
shown in the diagram above. The Earth has mass Me and radius Re. In this orbit, which has a radius of 2Re, the
satellites initially move with the same orbital speed vo but in opposite directions.
a. Calculate the orbital speed vo of the satellites in terms of G, Me, and Re.
b. Assume that the satellites collide head-on and stick together. In terms of vo find the speed v of the combination
immediately after the collision.
c. Calculate the total mechanical energy of the system immediately after the collision in terms of G, m, Me, and Re.
Assume that the gravitational potential energy of an object is defined to be zero at an infinite distance from the
Earth.
*1992M3. A spacecraft of mass 1,000 kilograms is in an elliptical orbit about the Earth, as shown above. At point A
the spacecraft is at a distance rA = 1.2 × 107 meters from the center of the Earth and its velocity, of magnitude
vA = 7.1 × 103 meters per second, is perpendicular to the line connecting the center of the Earth to the
spacecraft. The mass and radius of the Earth are ME = 6.0 × 1024 kilograms and rE = 6.4 × 106 meters,
respectively.
Determine each of the following for the spacecraft when it is at point A .
a. The total mechanical energy of the spacecraft, assuming that the gravitational potential energy is zero at an
infinite distance from the Earth.
b. The magnitude of the angular momentum of the spacecraft about the center of the Earth.
c.
Later the spacecraft is at point B on the exact opposite side of the orbit at a distance r B = 3.6 × 107 meters from
the center of the Earth.
Determine the speed vB of the spacecraft at point B.
d.
e.
Suppose that a different spacecraft is at point A, a distance r A = 1.2 × 107 meters from the center of the Earth.
Determine each of the following.
The speed of the spacecraft if it is in a circular orbit around the Earth
The minimum speed of the spacecraft at point A if it is to escape completely from the Earth
*1994M3 (modified) A satellite of mass m is in an elliptical orbit around the Earth, which has mass M e and radius
Re. The orbit varies from a closest approach of distance 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.
b.
c.
Determine the total energy of the satellite when it is at A.
What is the magnitude of the angular momentum of the satellite about the center of the Earth when it is at A?
Determine the velocity of the satellite as it passes point B in its orbit.
d.
e.
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.
Determine the speed of the satellite for this circular orbit.
Determine the work done by the rocket engine to effect this change.
*1995M3 (modified) Two stars, A and B. are in circular orbits of radii r a 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
b. The mass Mb of star B
c. The mass Ma of star A
d. Determine an expressions for the angular momentum of the system about the center of mass in terms of Ma,
Mb, ra, rb, T, and fundamental constants.
2007M2. 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 × 102 minutes = 7.08 × 103 s and orbital speed of 3.40 × 103
m/s . The mass of the GS is 930 kg and the radius of Mars is 3.43 × 106 m.
a.
b.
c.
d.
Calculate the radius of the GS orbit.
Calculate the mass of Mars.
Calculate the total mechanical energy of the GS in this orbit.
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
Justify your answer.
e.
_________ Less than
In fact, the orbit the GS entered was slightly elliptical with its closest approach to Mars at 3.71 × 105 m above
the surface and its furthest distance at 4.36 × 105 m above the surface. If the speed of the GS at closest approach
is 3.40 × 103 m/s, calculate the speed at the furthest point of the orbit.
2001M2. An explorer plans a mission to place a satellite into a circular orbit around the planet Jupiter, which has
mass MJ = 1.90 × 1027 kg and radius RJ = 7.14 × 107 m.
a. If the radius of the planned orbit is R, use Newton's laws to show each of the following.
i. The orbital speed of the planned satellite is given by
v
ii.
The period of the orbit is given by
T
b.
c.
GM J
R
4 2 R 3
GM J
The explorer wants the satellite's orbit to be synchronized with Jupiter's rotation. This requires an equatorial
orbit whose period equals Jupiter's rotation period of 9 hr 51 min = 3.55 × 104 s. Determine the required orbital
radius in meters.
Suppose that the injection of the satellite into orbit is less than perfect. For an injection velocity that differs from
the desired value in each of the following ways, sketch the resulting orbit on the figure. (J is the center of
Jupiter, the dashed circle is the desired orbit, and P is the injection point.) Also, describe the resulting orbit
qualitatively but specifically.
i. 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.
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.
2005M2. 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 MS of Saturn. Assume the orbits of these moons are circular.
a.
b.
c.
d.
e.
f.
Write an algebraic expression for the gravitational force between Saturn and one of its moons.
Use your expression from part (a) and 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.
Which quantities should be graphed to yield a straight line whose slope could be used to determine Saturn's
mass?
Complete the data table by calculating the two quantities to be graphed. Label the top of each column, including
units.
Plot the graph on the axes below. Label the axes with the variables used and appropriate numbers to indicate the
scale.
Using the graph, calculate a value for the mass of Saturn.