PROBLEM 13.86
A satellite describes an elliptic orbit of minimum altitude 606 km
above the surface of the earth. The semimajor and semiminor axes
are 17,440 km and 13,950 km, respectively. Knowing that the speed
of the satellite at Point C is 4.78 km/s, determine (a) the speed at
Point A, the perigee, (b) the speed at Point B, the apogee.
SOLUTION
6
6370 606 6976 km 6.976 10 m
A
r
= += = ×
PROBLEM 13.86 (Continued)
B
B
PROBLEM 13.87
While describing a circular orbit 200 mi above the earth a space vehicle launches a 6000–lb communications
satellite. Determine (a) the additional energy required to place the satellite in a geosynchronous orbit at an
altitude of 22,000 mi above the surface of the earth, (b) the energy required to place the satellite in the same
orbit by launching it from the surface of the earth, excluding the energy needed to overcome air resistance.
(A geosynchronous orbit is a circular orbit in which the satellite appears stationary with respect to the ground).
SOLUTION
300 50.1 10 ft lbE∆=× ⋅
PROBLEM 13.87 (Continued)
E
PROBLEM 13.88
How much energy per pound should be imparted to a satellite in order to place it in a circular orbit at an
altitude of (a) 400 mi, (b) 4000 mi?
SOLUTION
7960
W
W
PROBLEM 13.89
Knowing that the velocity of an experimental space probe fired
from the earth has a magnitude
32.5 Mm/h
A
v=
at Point A,
determine the speed of the probe as it passes through Point B.
SOLUTION
6
B
B
PROBLEM 13.90
A spacecraft is describing a circular orbit at an altitude of 1500 km above
the surface of the earth. As it passes through Point A, its speed is reduced by
40 percent and it enters an elliptic crash trajectory with the apogee at Point
A. Neglecting air resistance, determine the speed of the spacecraft when it
reaches the earth’s surface at Point B.
SOLUTION
PROBLEM 13.91
Observations show that a celestial body traveling at
6
1.2 10 mi/h×
appears to be describing about Point B a
circle of radius equal to 60 light years. Point B is suspected of being a very dense concentration of mass called
a black hole. Determine the ratio MB/MS of the mass at B to the mass of the sun. (The mass of the sun is
330,000 times the mass of the earth, and a light year is the distance traveled by light in one year at a velocity
of 186,300 mi/s.)
SOLUTION
21
sun
4.645 10
M
sun
M= ×
PROBLEM 13.92
(a) Show that, by setting
rRy= +
in the right-hand member of Eq. (13.17′) and expanding that member in a
power series in y/R, the expression in Eq. (13.16) for the potential energy Vg due to gravity is a first-order
approximation for the expression given in Eq. (13.17′). (b) Using the same expansion, derive a second-order
approximation for Vg.
g
PROBLEM 13.93
Collar A has a mass of 3 kg and is attached to a spring of constant
1200 N/m and of undeformed length equal to 0.5 m. The system is set
in motion with r = 0.3 m, v
θ
= 2 m/s, and vr = 0. Neglecting the mass
of the rod and the effect of friction, determine the radial and
transverse components of the velocity of the collar when r = 0.6 m.
SOLUTION
Let position 1 be the initial position.
1
0.3 m
r
=
θ
PROBLEM 13.94
Collar A has a mass of 3 kg and is attached to a spring of constant
1200 N/m and of undeformed length equal to 0.5 m. The system is set
in motion with r = 0.3 m, v
θ
= 2 m/s, and vr = 0. Neglecting the mass
of the rod and the effect of friction, determine (a) the maximum
distance between the origin and the collar, (b) the corresponding
speed. (Hint: Solve the equation obtained for r by trial and error.)
SOLUTION
Let position 1 be the initial position.
1
11 1
110
22
1
22
11
0.3 m
( ) 0, ( ) 2 m/s, 2 m/s
11
(3)(2) 6 J
22
11
(1200)( 0.2) 24 J
22
r
r
vv v
T mv
V kx
θ
=
= = =
= = =
= = −=
Let position 2 be when r is maximum.
2
() 0
r
v=
2
2
( 0.5)
11
m
m
rr
xr
=
= −
11 2 2
22
2
2
2
2
2
6 24 1.5( ) 600( 0.5)
0.6
30 (1.5) 600( 0.5)
0.54
( ) 600( 0.5) 30 0
m
m
m
mm
m
vr
r
r
fr r
r
θ
+= + −
= +−
= + − −=
PROBLEM 13.94 (Continued)
2
r
2
PROBLEM 13.95
A governor is designed so that the valve of negligible mass at D will
open once a vertical force greater than 20 lbs is exerted on it. In
initial testing of the device, the two 1 lb masses are at x = 1 in., and
are prevented from sliding along the rod by stops. Each mass is
connected to the valve by a 10 lb/in spring, which are both
unstretched at x = 1 in. The governor is rotating so that v1 = 30 ft/s
when the stops are removed. When the valve opens, determine the
position and velocity of the masses.
SOLUTION
1
2 21
1
22
2
0.4581 ft
30
0.1145 ft
lk
ll
k
s
= ⇒=
−
=
PROBLEM 13.95 (Continued)
2
2
4 0.3143 ft
228.04 ft/s 7.96 ft/s
rr
PROBLEM 13.96
A 1.5-lb ball that can slide on a horizontal frictionless surface is attached to a fixed
Point O by means of an elastic cord of constant
1 lb/in.k=
and undeformed length 2
ft. The ball is placed at Point A, 3 ft from O, and given an initial velocity
0
v
perpendicular to OA. Determine (a) the smallest allowable value of the initial speed
0
v
if the cord is not to become slack, (b) the closest distance d that the ball will come to
Point O if it is given half the initial speed found in part a.
SOLUTION
PROBLEM 13.96 (Continued)
2
1
PROBLEM 13.97
A 1.5-lb ball that can slide on a horizontal frictionless surface is attached to a fixed
Point O by means of an elastic cord of constant
1 lb/in.k=
and undeformed length 2
ft. The ball is placed at Point A, 3 ft from O, and given an initial velocity
0
v
perpendicular to OA, allowing the ball to come within a distance
9 in.d=
of Point O
after the cord has become slack. Determine (a) the initial speed
0
v
of the ball, (b) its
maximum speed.
SOLUTION
2
0.75
m
m
PROBLEM 13.98
Using the principles of conservation of energy and conservation of angular momentum, solve part a of Sample
Problem 12.14.
1
A
Conservation of energy:
Point A:
3
0
2 32
0
6
0
2 2 62
10.25 10 m/s
11
(10.25 10 )
22
( )(52.53 10 )(J)
A
A
A
v
T mv m
Tm
GMm
Vr
= ×
= = ×
= ×
= −
PROBLEM 13.98 (Continued)
1
max
PROBLEM 13.99
Solve sample Problem 13.11, assuming that the elastic cord is
replaced by a central force F of magnitude (80/r2) N directed
toward O.
SAMPLE PROBLEM 13.11 A sphere of mass
6m=
kg is
attached to an elastic cord of constant
100k=
N/m, which is
undeformed when the sphere is located at the origin O. Knowing
that the sphere may slide without friction on the horizontal surface
and that in the position shown its velocity vA has a magnitude of
20 m/s, determine (a) the maximum and minimum distances from
the sphere to the origin O, (b) the corresponding values of its
speed.
SOLUTION
m
m