Unlock document.
This document is partially blurred.
Unlock all pages and 1 million more documents.
Get Access
PROBLEM 12.75 (Continued)
22
00
n
n
ρ
2
0
r
PROBLEM 12.76
A particle of mass m is projected from Point A with an initial velocity v0
perpendicular to line OA and moves under a central force F along a
semicircular path of diameter OA. Observing that
0
cosrr
θ
=
and using
Eq. (12.25), show that the speed of the particle is
2
0
/cos .vv
θ
=
SOLUTION
0
cos
2
cos
θ
PROBLEM 12.77
For the particle of Problem 12.76, determine the tangential component
t
F
of the central force F along the tangent to the path of the particle for
(a)
0,
θ
=
(b)
45 .
θ
= °
PROBLEM 12.76 A particle of mass m is projected from Point A with an
initial velocity v0 perpendicular to line OA and moves under a central force
F along a semicircular path of diameter OA. Observing that
0
cosrr
θ
=
and using Eq. (12.25), show that the speed of the particle is
2
0
/cos .vv
θ
=
5
t
0
t
PROBLEM 12.78
Determine the mass of the earth knowing that the mean radius of the moon’s orbit about the earth is 238,910 mi
and that the moon requires 27.32 days to complete one full revolution about the earth.
SOLUTION
PROBLEM 12.79
Show that the radius r of the moon’s orbit can be determined from the radius R of the earth, the acceleration of
gravity g at the surface of the earth, and the time
τ
required for the moon to complete one full revolution about
the earth. Compute r knowing that
27.3
τ
=
days, giving the answer in both SI and U.S. customary units.
6
3960 mi 20.9088 10 ftR
= = ×
SI:
1/3
2 62 6 2
6
2
9.81 m/s (2.35872 10 s) (6.37 10 m) 382.81 10 m
4
r
π
× × ××
= = ×
3
PROBLEM 12.80
Communication satellites are placed in a geosynchronous orbit, i.e., in a circular orbit such that they complete
one full revolution about the earth in one sidereal day (23.934 h), and thus appear stationary with respect to
the ground. Determine (a) the altitude of these satellites above the surface of the earth, (b) the velocity with
which they describe their orbit. Give the answers in both SI and U.S. customary units.
SOLUTION
PROBLEM 12.80 (Continued)
6
138.334 10
×
PROBLEM 12.81
Show that the radius r of the orbit of a moon of a given planet can be determined from the radius R of the
planet, the acceleration of gravity at the surface of the planet, and the time
τ
required by the moon to complete
one full revolution about the planet. Determine the acceleration of gravity at the surface of the planet Jupiter
knowing that R = 71,492 km and that
τ
= 3.551 days and r = 670.9 × 103 km for its moon Europa.
SOLUTION
Mm
Jupiter Earth
PROBLEM 12.82
The orbit of the planet Venus is nearly circular with an orbital velocity of 126.5
3
10×
km/h. Knowing that
the mean distance from the center of the sun to the center of Venus is 108
6
10×
km and that the radius of the
sun is
3
695 10×
km, determine (a) the mass of the sun, (b) the acceleration of gravity at the surface of the
sun.
SOLUTION
2 62
(695.5 10 )
PROBLEM 12.83
A satellite is placed into a circular orbit about the planet Saturn at an altitude of 2100 mi. The satellite
describes its orbit with a velocity of
3
54.7 10 mi/h.×
Knowing that the radius of the orbit about Saturn and the
periodic time of Atlas, one of Saturn’s moons, are
3
85.54 10 mi×
and 0.6017 days, respectively, determine
(a) the radius of Saturn, (b) the mass of Saturn. (The periodic time of a satellite is the time it requires to
complete one full revolution about the planet.)
SOLUTION
2A
r
π
PROBLEM 12.84
The periodic time (see Prob. 12.83) of an earth satellite in a circular polar
orbit is 120 minutes. Determine (a) the altitude h of the satellite, (b) the time
during which the satellite is above the horizon for an observer located at the
north pole.
SOLUTION
PROBLEM 12.85
A 500 kg spacecraft first is placed into a circular orbit about the earth at an altitude of 4500 km and then is
transferred to a circular orbit about the moon. Knowing that the mass of the moon is 0.01230 times the mass
of the earth and that the radius of the moon is 1737 km, determine (a) the gravitational force exerted on the
spacecraft as it was orbiting the earth, (b) the required radius of the orbit of the spacecraft about the moon if
the periodic times (see Problem 12.83) of the two orbits are to be equal, (c) the acceleration of gravity at the
surface of the moon.
SOLUTION
6
M
M
PROBLEM 12.85 (Continued)
2[Eq.(12.29)]GM gR=
moon earth
6
PROBLEM 12.86
A space vehicle is in a circular orbit of 2200-km radius around the moon.
To transfer it to a smaller circular orbit of 2080-km radius, the vehicle is
first placed on an elliptic path AB by reducing its speed by 26.3 m/s as it
passes through A. Knowing that the mass of the moon is 73.49
21
10 kg,×
determine (a) the speed of the vehicle as it approaches B on the elliptic
path, (b) the amount by which its speed should be reduced as it
approaches B to insert it into the smaller circular orbit.
SOLUTION
2
nn
v
Eq. (12.28):
2
Mm
FG
r
=
Then
2
2
Mm v
Gm
r
r=
or
2GM
vr
=
Then
12 3 2 21
2
circ 3
66.73 10 m /kg s 73.49 10 kg
() 2200 10 m
A
v
−
× ⋅× ×
=×
or
circ
( ) 1493.0 m/s
A
v=
and
12 3 2 21
2
circ 3
66.73 10 m /kg s 73.49 10 kg
() 2080 10 m
B
v
−
× ⋅× ×
=×
or
circ
( ) 1535.5 m/s
B
v=
(a) We have
circ
() ()
(1493.0 26.3) m/s
1466.7 m/s
A TR A A
vv v= +∆
= −
=
Conservation of angular momentum requires that
() ()
A A TR B B TR
rmv rmv=
or
2200 km
( ) 1466.7 m/s
2080 km
1551.3 m/s
B TR
v= ×
=
or
( ) 1551 m/s
B TR
v=
(b) Now
circ
( ) ()
B B TR B
v vv= +∆
or
(1535.5 1551.3) m/s
B
v∆= −
or
15.8 m/s
B
v∆=−
PROBLEM 12.87
As a first approximation to the analysis of a space flight from the
earth to Mars, assume the orbits of the earth and Mars are
circular and coplanar. The mean distances from the sun to the
earth and to Mars are
6
149.6 10×
km and
6
227.8 10×
km,
respectively. To place the spacecraft into an elliptical transfer
orbit at point A, its speed is increased over a short interval of
time to
A
v
which is 2.94 km/s faster than the earth’s orbital
speed. When the spacecraft reaches point B on the elliptical
transfer orbit, its speed
B
v
is increased to the orbital speed of
Mars. Knowing that the mass of the sun is
3
332.8 10×
times the
mass of the earth, determine the increase in speed required at B.
SOLUTION
2
2 6 12 3 2
B
PROBLEM 12.88
To place a communications satellite into a geosynchronous orbit (see
Problem 12.80) at an altitude of 22,240 mi above the surface of the
earth, the satellite first is released from a space shuttle, which is in a
circular orbit at an altitude of 185 mi, and then is propelled by an upper-
stage booster to its final altitude. As the satellite passes through A, the
booster’s motor is fired to insert the satellite into an elliptic transfer
orbit. The booster is again fired at B to insert the satellite into a
geosynchronous orbit. Knowing that the second firing increases the
speed of the satellite by 4810 ft/s, determine (a) the speed of the satellite
as it approaches B on the elliptic transfer orbit, (b) the increase in speed
resulting from the first firing at A.
SOLUTION
6
15
6
3
14.077 10
138.336 10
10.088 10 ft/s
B
×
=×
= ×
PROBLEM 12.88 (Continued)
A
PROBLEM 12.89
A space vehicle is in a circular orbit of 1400-mi radius around the moon.
To transfer to a smaller orbit of 1300-mi radius, the vehicle is first placed
in an elliptic path AB by reducing its speed by 86 ft/s as it passes through
A. Knowing that the mass of the moon is
21 2
5.03 10 lb s /ft,×⋅
determine
(a) the speed of the vehicle as it approaches B on the elliptic path, (b) the
amount by which its speed should be reduced as it approaches B to insert
it into the smaller circular orbit.
SOLUTION
GM
B
PROBLEM 12.90
A 1 kg collar can slide on a horizontal rod, which is free to
rotate about a vertical shaft. The collar is initially held at A by a
cord attached to the shaft. A spring of constant 30 N/m is
attached to the collar and to the shaft and is undeformed when
the collar is at A. As the rod rotates at the rate
16 rad/s,
θ
=
the
cord is cut and the collar moves out along the rod. Neglecting
friction and the mass of the rod, determine (a) the radial and
transverse components of the acceleration of the collar at A,
(b) the acceleration of the collar relative to the rod at A, (c) the
transverse component of the velocity of the collar at B.
SOLUTION
B
θ
PROBLEM 12.91
A 1-lb ball A and a 2-lb ball B are mounted on a horizontal rod
which rotates freely about a vertical shaft. The balls are held in
the positions shown by pins. The pin holding B is suddenly
removed and the ball moves to position C as the rod rotates.
Neglecting friction and the mass of the rod and knowing that
the initial speed of A is
8
A
v=
ft/s, determine (a) the radial and
transverse components of the acceleration of ball B
immediately after the pin is removed, (b) the acceleration of
ball B relative to the rod at that instant, (c) the speed of ball A
after ball B has reached the stop at C.
SOLUTION
B
θ
