Download 3 - WordPress.com

Problem Set 13: Gravitation
AP Physics C Supplementary Problems
1.
How far from the earth must a space probe be along a line toward the sun so that
the sun’s gravitational pull balances the earth’s?
2.
Two concentric shells of uniform density having masses M1 and M2 are situated
as shown below. Find the force on a particle of mass m when the particle is
located at (a) r = a, (b) r = b, and (c) r = c. The distance r is measured from the
center of the shells.
M2
M1
c
a
b
3.
Show that, at the bottom of a vertical mine shaft dug to depth D, the measured
value of g will be
g = gs 1− D ,
R
gs being the surface value. Assume that the earth is a uniform sphere of radius R.
(
4.
)
A spherical hollow is made in a lead sphere of radius R, such that its surface
touches the outside surface of the lead sphere and passes through its center. The
mass of the sphere before hollowing was M. With what force, according to
Newton’s law of gravitation, will the lead sphere attract a small sphere of mass m,
which lies at a distance d from the center of the lead sphere on the straight line
connecting the centers of the spheres and of the hollow?
m
R
d
Problems selected from Halliday, D., & Resnick, R. (1993). Fundamentals of Physics (4th ed.). New
York: John Wiley & Sons, Inc.
Gravitation
2
5.
At what altitude above the earth’s surface would the gravitational acceleration be
4.9 m/s2?
6.
The fastest possible rate of rotation of a planet is that for which the gravitational
force on material at the equator barely provides the centripetal force needed for
the rotation. Show then that the corresponding shortest period of rotation is given
by
T = 3π
Gρ
where ρ is the density of the planet, assumed homogeneous.
7.
The sun, mass 2.0 x 1030 kg, is revolving about the center of the Milky Way
galaxy, which is 2.2 x 1020 m away. It completes one revolution every 2.5 x 108
years. Estimate roughly the number of stars in the Milky Way. Assume for
simplicity that the stars are distributed with spherical symmetry about the galactic
center and that our sun is essentially at the galactic edge.
8.
Masses of 200 and 800 g are 12 cm apart. (a) Find the gravitational force on a
100 g object situated at a point on the line joining the masses 4.0 cm from the 200
g mass. (b) How much work is needed to move this object to a point 4.0 cm from
the 800 g mass along the line of centers?
9.
A projectile is fired vertically from the earth’s surface with an initial speed of 10
km/s. Neglecting atmospheric friction, how far above the surface of the earth will
it go?
10.
Two particles with masses m and M are initially at rest an infinite distance apart.
Show that at any instant their relative velocity of approach attributable to
gravitational attraction is 2G ( M + m) , where d is their separation at that
d
instant. (Hint: Use conservation of energy and conservation of linear
momentum.)
11.
The planet Mars has a satellite, Phobos, which travels in an orbit of radius 9.4 x
106 m with a period of 7 hr 39 min. Calculate the mass of Mars from this
information.
12.
(a) With what horizontal speed must a satellite be projected at 160 km above the
surface of the earth so that it will have a circular orbit about the earth? (b) What
will be the period of revolution?
13.
Consider an artificial satellite in a circular orbit about the earth. State how the
following properties of the satellite vary with the radius r of its orbit: (a) period,
(b) kinetic energy, (c) angular momentum, and (d) speed.
Gravitation
3
Answers:
1.
2.59 x 108 m
G ( M 1 + M 2 )m
2.
a)
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
a
2
b)
GM 1 m
b)
-5.0 x 10-11 J
b)
87.4 min
b)
1/r
c)
b2
0
proof
⎛
⎜
1
GMm ⎜
1−
⎜
2
2
d ⎜
R ⎞
⎛
⎜ 8⎜1 − 2d ⎟
⎠
⎝
⎝
6
2.65 x 10 m
proof
5.1 x 1010 stars
a)
0N
2.5 x 107 m
proof
6.48 x 1023 kg
a)
7820 m/s
a)
d)
r3
1
r
⎞
⎟
⎟
⎟
⎟
⎟
⎠
c)
r