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
where
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
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
10 rad/s
2
.
Knowing that when
º 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.
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 b20 in.
Neglecting friction and assuming that 15 rad/s
and
2
250 rad/s
for the position
20 ,
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.
PROBLEM 12.71
The two blocks are released from rest when r 0.8 m and 30 .
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.
yBBB BB
Adding Eq. (1) to Eq. (2) to eliminate T,
sec
BAA BB
mg ma ma
(4)
Radial and transverse components of
.
A
a
Use either the scalar product of vectors or the triangle construction shown,
being careful to note the positive directions of the components.
2
cos
rAr A
rr a a
ae
(5)
Noting that initially
0,
using Eq. (1) to eliminate
,r
and changing
signs gives
cos
BA
aa
(6)
Substituting Eq. (6) into Eq. (4) and solving for
,
A
a
2
(25) (9.81) 5.48 m/s
sec cos 20sec30 25cos30
B
AAB
mg
amm
From Eq. (6),
2
5.48cos30 4.75 m/s
B
a
(a) From Eq. (2), (20)(5.48)sec30 126.6T 126.6 NT
(b) Acceleration of block A.
2
5.48 m/s
A
a
(c) Acceleration of block B.
2
4.75 m/s
B
a
PROBLEM 12.72
The velocity of block A is 2 m/s to the right at the instant when
0.8 mr and 30 .
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.
Using Eq. (1) to eliminate r
and changing signs gives
2
cos
BA
aa r
(6)
PROBLEM 12.72 (Continued)
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 010 rad/s
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.
PROBLEM 12.73* (Continued)
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/cos2rr
and using Eq. (12.25), express
the radial and transverse components of the velocity v of the particle as
functions of .
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/cos2rr
and using
Eq. (12.25), express the radial and transverse components of the velocity
v of the particle as functions of
.