PROBLEM 12.121
Show that the angular momentum per unit mass h of a satellite describing an elliptic orbit of semimajor axis a
and eccentricity about a planet of mass M can be expressed as
SOLUTION
2
(1 )h GMa
ε
= −
ε
2
(1 )h GMa
ε
= −
1(1 cos )
GM
2
ε
PROBLEM 12.122
In the braking test of a sports car its velocity is reduced from 70 mi/h to zero in a distance of 170 ft with
slipping impending. Knowing that the coefficient of kinetic friction is 80 percent of the coefficient of static
friction, determine (a) the coefficient of static friction, (b) the stopping distance for the same initial velocity if
the car skids. Ignore air resistance and rolling resistance.
0
PROBLEM 12.123
A bucket is attached to a rope of length L = 1.2 m and is made to revolve in a
horizontal circle. Drops of water leaking from the bucket fall and strike the floor
along the perimeter of a circle of radius a. Determine the radius a when
SOLUTION
30 .
θ
= °
Copyright © McGraw–Hill Education. Permission required for reproduction or display.
PROBLEM 12.123 (Continued)
Projection on horizontal floor (uniform motion)
Radius of circle:
Note: The drop travels in a vertical plane parallel to the yz plane.
00
00
( ) sin 30 0, 0.6 m
( ) 0 1.843(0.527) 0.971 m
z
z
x x vt L x
z z vt
= + = °+ =
=+=+ =
22
a xz= +
22
(0.6) (0.971)a= +
1.141 ma=
PROBLEM 12.124
A 12-lb block B rests as shown on the upper surface of a 30-lb wedge A.
Neglecting friction, determine immediately after the system is released from
rest (a) the acceleration of A, (b) the acceleration of B relative to A.
SOLUTION
/
BA
PROBLEM 12.125
A 500–lb crate B is suspended from a cable attached to a 40–lb trolley A
which rides on an inclined I-beam as shown. Knowing that at the instant
shown the trolley has an acceleration of 1.2 ft/s2 up and to the right,
determine (a) the acceleration of B relative to A, (b) the tension in cable CD.
SOLUTION
PROBLEM 12.126
The roller–coaster track shown is contained in a vertical plane.
The portion of track between A and B is straight and
horizontal, while the portions to the left of A and to the right
of B have radii of curvature as indicated. A car is traveling at a
speed of 72 km/h when the brakes are suddenly applied,
causing the wheels of the car to slide on the track
Determine the initial deceleration of the car if the brakes are
applied as the car (a) has almost reached A, (b) is traveling
between A and B, (c) has just passed B.
SOLUTION
( 0.25).
k
µ
=
y y RF n
PROBLEM 12.126 (Continued)
t
PROBLEM 12.127
The parasailing system shown uses a
winch to pull the rider in towards the
boat, which is travelling with a
constant velocity. During the interval
when
θ
is between 20º and 40º, (where t
= 0 at
θ
= 20º) the angle increases at the
constant rate of 2 º/s. During this time,
the length of the rope is defined by the
relationship
3/2
1
125 3
rt= −
, where r
and t are expressed in meters and
seconds, respectively. At the instant
when the rope makes a 30 degree angle
with the water, the tension in the rope is
18 kN. At this instant, what is the
magnitude and direction of the force of
the parasail on the 75 kg parasailor?
2
cos sin (1)
P
P
F T mg r r
b θθ
= + +−
sin cos 2 (2)
P
P
F mg r r
b θθ θ
= ++
PROBLEM 12.127 (Continued)
P
P
PROBLEM 12.128
A small 200–g collar C can slide on a semicircular rod which is made to rotate about
the vertical AB at the constant rate of 6 rad/s. Determine the minimum required value
of the coefficient of static friction between the collar and the rod if the collar is not to
slide when (a) (b) (c) Indicate in each case the direction
of the impending motion.
SOLUTION
90 ,
θ
= °
90 ,
θ
= °
75 ,
θ
= °
45 .
θ
= °
PROBLEM 12.128 (Continued)
1 2.2018sin 75 tan 75
s
+ °°
PROBLEM 12.128 (Continued)
PROBLEM 12.129
Telemetry technology is used to quantify kinematic values of a
200-kg roller coaster cart as it passes overhead. According to
the system,
At this instant, determine
(a) the normal force between the cart and the track, (b) the
radius of curvature of the track.
25 m, 10 m/s,rr= = −
2
2 m/s , 90 ,r
θ
=−=°
0.4 rad/s,
θ
= −
θ
=
2
0.32 rad/s .−
PROBLEM 12.130
The radius of the orbit of a moon of a given planet is equal to twice the radius of that planet. Denoting
by
ρ
the mean density of the planet, show that the time required by the moon to complete one full revolution
about the planet is where G is the constant of gravitation.
1/2
(24 / ) ,G
πρ
PROBLEM 12.131
At engine burnout on a mission, a shuttle had reached Point A at an
altitude of 40 mi above the surface of the earth and had a horizontal
velocity v0. Knowing that its first orbit was elliptic and that the shuttle
was transferred to a circular orbit as it passed through Point B at an
altitude of 170 mi, determine (a) the time needed for the shuttle to travel
from A to B on its original elliptic orbit, (b) the periodic time of the
shuttle on its final circular orbit.
SOLUTION
62
PROBLEM 12.131 (Continued)
PROBLEM 12.132
A space probe in a low earth orbit is inserted into an elliptic
transfer orbit to the planet Venus. Knowing that the mass of the
sun is times the mass of the earth and assuming that
the probe is subjected only to the gravitational attraction of the
sun, determine the value of which defines the relative position
of Venus with respect to the earth at the time the probe is inserted
into the transfer orbit.
SOLUTION
3
332.8 10×
,
φ
v
r
PROBLEM 12.132 (Continued)
PROBLEM 12.133*
Disk A rotates in a horizontal plane about a vertical axis at the
constant rate rad/s. Slider B has mass 1 kg and moves in a
frictionless slot cut in the disk. The slider is attached to a spring of
constant k, which is undeformed when Knowing that the slider
is released with no radial velocity in the position
determine the position of the slider and the horizontal force exerted
on it by the disk at for (a) (b)
SOLUTION
0
10
θ
=
0.r=
500 mm,r=
0.1 st=
100 N/m,k=
200 N/m.k=