Download 1992M3. A spacecraft of mass 1,000 kilograms is in an elliptical orbit

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 r A = 1.2 x 107 meters from the center of the Earth and its velocity, of
magnitude VA = 7.1 x 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 X 1024 kilograms and rE = 6.4 X 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.
Later the spacecraft is at point B on the exact opposite side of the orbit at a distance r B = 3.6 X 107
meters from the center of the Earth.
c. Determine the speed vB of the spacecraft at point B.
Suppose that a different spacecraft is at point A, a distance rA = 1.2 X 107 meters from the center of the
Earth. Determine each of the following.
d. The speed of the spacecraft if it is in a circular orbit around the Earth
e. The minimum speed of the spacecraft at point A if it is to escape completely from the Earth
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.
1984M3. A small body of mass m located near the Earth’s surface falls from rest in the Earth's
gravitational
field. Acting on the body is a resistive force of magnitude kmv, where k is a constant and v is the speed
of the body.
a. On the diagram below, draw and identify all of the forces acting on the body as it falls.
b. Write the differential equation that represents Newton's second law for this situation.
c. Determine the terminal speed vT of the body.
d. Integrate the differential equation once to obtain an expression for the speed v as a function of
time t. Use the condition that v = 0 when t= 0.
e.
On the axes provided below, draw a graph of the speed v as a function of time t.
1994M3. 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 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, M e, 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.
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.
1995M3. 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 r a, 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
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.
1996M1. A thin, flexible metal plate attached at one end to a platform, as shown above, can be used to
measure mass. When the free end of the plate is pulled down and released, it vibrates in simple harmonic
motion with a period that depends on the mass attached to the plate. To calibrate the force constant, objects
of known mass are attached to the plate and the plate is vibrated, obtaining the data shown below.
a. Fill in the blanks in the data table.
b.
c.
On the graph below, plot T 2 versus mass. Draw on the graph the line that is your estimate of the best
straight-line fit to the data points.
An object whose mass is not known is vibrated on the plate, and the average time for ten vibrations is
measured to be 16.1 s. From your graph, determine the mass of the object. Write your answer with a
reasonable number of significant digits.
d. Explain how one could determine the force constant of the metal plate.
e. Can this device be used to measure mass aboard the space shuttle Columbia as it orbits the Earth?
Explain briefly.
If Columbia is orbiting at 0.3 x 10 6 m above the Earth's surface, what is the acceleration of Columbia
due to the Earth's gravity? (Radius of Earth = 6.4 x 10 6 m, mass of Earth = 6.0 x 1024 kg)
g. Since the answer to part (f) is not zero, briefly explain why objects aboard the orbiting Columbia seem
weightless.
f.
1999 M2 A spherical, nonrotating planet has a radius R and a uniform density  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. Show that the magnitude of the force on the ball at a distance r < R from the center of
the planet is given by F = -Cr. where C = 4/3(Gm).
b. On the axes below, sketch the force F on the ball as a function of distance r from the
center of the planet.
The ball is dropped into the tunnel from rest at point P at the planet's surface.
c. Determine the work done by gravity as the ball moves from the surface to the center
of the planet.
d. Determine the speed of the ball when it reaches the center of the planet.
e. Fully describe the subsequent motion of the ball from the time it reaches the center of
the planet.
f.
Write an equation that could be used to calculate the time it takes the ball to move from point P to the
center of the planet. It is not necessary to solve this equation.
2001M2. An explorer plans a mission to place a satellite into a circular orbit around the
planet Jupiter, which has mass MJ = 1.90 x 1027 kg and radius RJ = 7.14 x 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.
GM J
R
The period of the orbit is given by
T
4 2 R 3
GM J
b. 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 x 104 s. Determine the required orbital radius in meters.
c.
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.