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PROBLEM 12.62 (Continued)
PROBLEM 12.63
Knowing that the coefficients of friction between the component I and member BC of the mechanism of
Problem 12.62 are
0.35
s
µ
=
and
0.25,
k
µ
=
determine (a) the maximum allowable constant speed
B
v
if the
component is not to slide on BC while being transferred, (b) the values of
θ
for which sliding is impending.
SOLUTION
2
B
v
W
PROBLEM 12.63 (Continued)
n
PROBLEM 12.63 (Continued)
B
PROBLEM 12.64
A small 250-g collar C can slide on a semicircular rod which is made to rotate about
the vertical AB at a constant rate of 7.5 rad/s. Determine the three values of
θ
for
which the collar will not slide on the rod, assuming no friction between the collar
and the rod.
2
sin ( sin )
cos
mg mr
θθω
θ
=
(1)
PROBLEM 12.65
A small 250-g collar C can slide on a semicircular rod which is made to rotate about
the vertical AB at a constant rate of 7.5 rad/s. Knowing that the coefficients of
friction are
µ
s = 0.25 and
µ
k = 0.20, indicate whether the collar will slide on the rod if
it is released in the position corresponding to (a)
θ
= 75°, (b)
θ
= 40°. Also,
determine the magnitude and direction of the friction force exerted on the collar
immediately after release.
SOLUTION
PROBLEM 12.65 (Continued)
2
PROBLEM 12.66
An advanced spatial disorientation trainer allows the cab to rotate around
multiple axes as well as extend inwards and outwards. It can be used to
simulate driving, fixed wing aircraft flying, and helicopter maneuvering. In
one training scenario, the trainer rotates and translates in the horizontal plane
where the location of the pilot is defined by the relationships
10 2cos 3
rt
π
= +
and
( )
2
0.1 2tt
θ
= −
, where r,
θ
, and t are expressed
in feet, radians, and seconds, respectively. Knowing that the pilot has a mass
of 175 lbs, (a) find the magnitude of the resulting force acting on the pilot at
t= 5 s (b) plot the magnitudes of the radial and transverse components of the
force exerted on the pilot from 0 to 10 seconds.
SOLUTION
π
zz
∑
PROBLEM 12.66 (Continued)
rz
θ
(b) Plot of Fr and Fθ from t=0 to t=10 s:
0 1 2 3 4 5 6 7 8 9 10
-800
-700
-600
-500
-400
-300
-200
-100
0
100
Radial and Transverse Components of Force for t=0 to 10 seconds
Time, (sec)
Force, (lbs)
Radial Component
Transverse Component
PROBLEM 12.67
An advanced spatial disorientation trainer is programmed to only rotate
and translate in the horizontal plane. The pilot’s location is defined by the
relationships
( )
81 t
re
−
= −
and
2 / sin 2t
π
θπ
=
, where r, θ, and t
are expressed in feet, radians, and seconds, respectively. Determine the
radial and transverse components of the force exerted on the 175 lb pilot at
t = 3 s.
SOLUTION
t
−
θ
PROBLEM 12.68
The 3-kg collar B slides on the frictionless arm
.AA′
The arm is
attached to drum D and rotates about O in a horizontal plane at the
rate
0.75 ,t
θ
=
where
θ
and t are expressed in rad/s and seconds,
respectively. As the arm-drum assembly rotates, a mechanism
within the drum releases cord so that the collar moves outward from
O with a constant speed of 0.5 m/s. Knowing that at t = 0, r = 0,
determine the time at which the tension in the cord is equal to the
magnitude of the horizontal force exerted on B by arm
.AA′
SOLUTION
Kinematics
dr r
PROBLEM 12.69
A 0.5-kg block B slides without friction inside a slot cut in arm OA which rotates in
a vertical plane. The rod has a constant angular acceleration
θ
= 10 rad/s2.
Knowing that when
θ
= 45º and r = 0.8 m the velocity of the block is zero,
determine at this instant, (a) the force exerted on the block by the arm, (b) the
relative acceleration of the block with respect to the arm.
SOLUTION
PROBLEM 12.70
Pin B weighs 4 oz and is free to slide in a horizontal plane along the
rotating arm OC and along the circular slot DE of radius b
20 in.=
Neglecting friction and assuming that
15 rad/s
θ
=
and
2
250 rad/s
θ
=
for the position
20 ,
θ
= °
determine for that position
(a) the radial and transverse components of the resultant force
exerted on pin B, (b) the forces P and Q exerted on pin B,
respectively, by rod OC and the wall of slot DE.
SOLUTION
Kinematics.
PROBLEM 12.71
The two blocks are released from rest when r = 0.8 m and
30 .
θ
= °
Neglecting the mass of the pulley and the effect of friction in the
pulley and between block A and the horizontal surface, determine
(a) the initial tension in the cable, (b) the initial acceleration of block
A, (c) the initial acceleration of block B.
SOLUTION
B
PROBLEM 12.72
The velocity of block A is 2 m/s to the right at the instant when
0.8 mr=
and
30 .
θ
= °
Neglecting the mass of the pulley and the
effect of friction in the pulley and between block A and the horizontal
surface, determine, at this instant, (a) the tension in the cable, (b) the
acceleration of block A, (c) the acceleration of block B.
SOLUTION
BA
PROBLEM 12.72 (Continued)
B=a
PROBLEM 12.73*
Slider C has a weight of 0.5 lb and may move in a slot cut in arm
AB, which rotates at the constant rate
0
10 rad/s
θ
=
in a horizontal
plane. The slider is attached to a spring of constant k = 2.5 lb/ft,
which is unstretched when r = 0. Knowing that the slider is
released from rest with no radial velocity in the position r = 18 in.
and neglecting friction, determine for the position r = 12 in. (a) the
radial and transverse components of the velocity of the slider,
(b) the radial and transverse components of its acceleration, (c) the
horizontal force exerted on the slider by arm AB.
SOLUTION
PROBLEM 12.73* (Continued)
θθ
θ
PROBLEM 12.74
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 directed
away from the center of force O. Knowing that the particle follows a path
defined by the equation
0
/ cos 2rr
θ
=
and using Eq. (12.25), express
the radial and transverse components of the velocity v of the particle as
functions of
.
θ
SOLUTION
0
0
θ
PROBLEM 12.75
For the particle of Problem 12.74, show (a) that the velocity of the
particle and the central force F are proportional to the distance r from
the particle to the center of force O, (b) that the radius of curvature of
the path is proportional to r3.
PROBLEM 12.74 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 directed away from the center of force O. Knowing that the
particle follows a path defined by the equation
0/ cos 2rr
θ
=
and using
Eq. (12.25), express the radial and transverse components of the velocity
v of the particle as functions of
θ
.
SOLUTION
