PROBLEM 12.36
A 450–g tetherball A is moving along a horizontal circular path at a
constant speed of 4 m/s. Determine (a) the angle
θ
that the cord
forms with pole BC, (b) the tension in the cord.
PROBLEM 12.37
During a hammer thrower’s practice swings, the 7.1-kg head
A of the hammer revolves at a constant speed v in a horizontal
circle as shown. If
0.93 m
ρ
=
and
60 ,
θ
= °
determine
(a) the tension in wire BC, (b) the speed of the hammer’s
head.
SOLUTION
2
A
v
A
PROBLEM 12.38
Human centrifuges are often used to simulate different acceleration
levels for pilots. When aerospace physiologists say that a pilot is
pulling 9g’s, they mean that the resultant normal force on the pilot from
the bottom of the seat is nine times their weight. Knowing that the
centrifuge starts from rest and has a constant angular acceleration of
1.5 RPM per second until the pilot is pulling 9g’s and then continues
with a constant angular velocity, determine (a) how long it will take for
the pilot to reach 9g’s (b) the angle θ of the normal force once the pilot
reaches 9 g’s. Assume that the force parallel to the seat is zero.
SOLUTION
2
PROBLEM 12.39
A single wire ACB passes through a ring at C attached to a sphere which
revolves at a constant speed v in the horizontal circle shown. Knowing that
the tension is the same in both portions of the wire, determine the speed v.
PROBLEM 12.40*
Two wires AC and BC are tied at C to a sphere which revolves at a
constant speed v in the horizontal circle shown. Determine the range of
the allowable values of v if both wires are to remain taut and if the
tension in either of the wires is not to exceed 60 N.
SOLUTION
From the solution of Problem 12.39, we find that both wires remain taut for
3.01 m/s 3.96 m/sv≤≤
To determine the values of v for which the tension in either wire will not exceed 60 N, we recall
Eqs. (1) and (2) from Problem 12.39:
2
mv
AC
1
PROBLEM 12.40* (Continued)
2
PROBLEM 12.41
A 1–kg sphere is at rest relative to a parabolic dish which rotates at a
constant rate about a vertical axis. Neglecting friction and knowing
that r
=
1 m, determine (a) the speed v of the sphere, (b) the
magnitude of the normal force exerted by the sphere on the inclined
surface of the dish.
SOLUTION
2
r dy
PROBLEM 12.42*
As part of an outdoor display, a 12-lb model C of the earth is attached to wires AC and
BC and revolves at a constant speed v in the horizontal circle shown. Determine the
range of the allowable values of v if both wires are to remain taut and if the tension in
either of the wires is not to exceed 26 lb.
max
[( ) 8.1964 lb]
CB
T
=
PROBLEM 12.42* (Continued)
CA CB
C
PROBLEM 12.43*
The 1.2-lb flyballs of a centrifugal governor revolve at a constant speed v
in the horizontal circle of 6–in. radius shown. Neglecting the weights of
links AB, BC, AD, and DE and requiring that the links support only tensile
forces, determine the range of the allowable values of v so that the
magnitudes of the forces in the links do not exceed 17 lb.
SOLUTION
2
v
Case 1: is maximum.
DA
T
DA
PROBLEM 12.43* (Continued)
max
( ) 16.9443 lb
DA
T
=
max
Case 2: is minimum.
DA
T
PROBLEM 12.44
A 130-lb wrecking ball B is attached to a
45-ft-long
steel cable
AB and swings in the vertical arc shown. Determine the tension
in the cable (a) at the top C of the swing, (b) at the bottom D of
the swing, where the speed of B is 13.2 ft/s.
SOLUTION
BA
BA
PROBLEM 12.45
During a high-speed chase, a 2400-lb sports car traveling at a
speed of 100 mi/h just loses contact with the road as it reaches
the crest A of a hill. (a) Determine the radius of curvature
ρ
of
the vertical profile of the road at A. (b) Using the value of
ρ
found in part a, determine the force exerted on a 160–lb driver
by the seat of his 3100-lb car as the car, traveling at a constant
speed of 50 mi/h, passes through A.
SOLUTION
PROBLEM 12.46
An airline pilot climbs to a new flight level along the path shown.
Knowing that the speed of the airplane decreases at a constant rate from
180 m/s at point A to 160 m/s at point C, determine the magnitude of the
abrupt change in the force exerted on a 90-kg passenger as the airplane
passes point B.
PROBLEM 12.47
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
( 0.20).
k
µ
=
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
2
:
v
F ma N mg m
Σ= − =
tk
m
t
PROBLEM 12.47 (Continued)
2
:
mv
F ma mg N
Σ= −=
22
(20)
0.20 9.81 45
tk
Fv
ag
m
µρ
== −= −
2
0.1842 m/s
t
a=
PROBLEM 12.48
A spherical–cap governor is fixed to a vertical shaft that rotates with angular
velocity
ω
. When the string-supported clapper of mass m touches the cap, a
cutoff switch is operated electrically to reduce the speed of the shaft.
Knowing that the radius of the clapper is small relative to the cap, determine
the minimum angular speed at which the cutoff switch operates.
SOLUTION
win
PROBLEM 12.49
A series of small packages, each with a mass of 0.5 kg, are discharged from a
conveyor belt as shown. Knowing that the coefficient of static friction between
each package and the conveyor belt is 0.4, determine (a) the force exerted by the
belt on a package just after it has passed Point A, (b) the angle
θ
defining the
Point B where the packages first slip relative to the belt.
SOLUTION
Assume package does not slip.
θ
sin ( cos )
4
sin cos 0.40 cos 9.81
sn
n
s
mg mg ma
a
g
θµ θ
θµ θ θ
= −
= −= −
PROBLEM 12.49 (Continued)
Squaring and using trigonometic identities,
22
ss
µµ
PROBLEM 12.50
A 54-kg pilot flies a jet trainer in a half vertical loop of 1200–m
radius so that the speed of the trainer decreases at a constant rate.
Knowing that the pilot’s apparent weights at Points A and C are 1680
N and 350 N, respectively, determine the force exerted on her by the
seat of the trainer when the trainer is at Point B.
SOLUTION
22
2
22,555 m /s
=