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Unformatted text preview: I. 968 Kinematics of Rigid Bodies 15.122 Arm AB has a constant angular velocity of l6 rad/s countérul wise. At the instant when 0 = 0, determine the acceleratioxgb.
collar D, (b) of the midpoint G of bar BD. ' Fig. ”5.122, P15.123, and P15.124 15.123 Arm AB has a constant angular velocity of 16 rad/s counte
clockwise. At the instant when 6 = 90°, determine the acceleraﬁ..
(a) of collar D, (b) of the midpoint G of bar BD. 15.124 Arm AB has a constant angular velocity of 16 rad/s coun't clockwise. At the instant when 0 = 60°, determine the accelerau...
of collar D. 15.125 Knowing that crank AB rotates about point A with a constant angular velocity of 900 rpm clockwise, determine the acceleratio I
of the piston P when 9 = 60°. ' 15.126 Knowing that crank AB rotates about point A with a const‘. 2 angular velocity of 900 rpm clockwise, determine the accelera’tio I
Fig. P15.'l25 and P15.‘l26 of the piston P when 0 = 120°. 15.127 Knowing that at the instant shown rod AB has zero angular acce
eration and an angular velocity of 15 rad/s counterclockwise, deter : _ mine (a) the angular acceleration of arm DE, (19) the acceleratio
' of point D. Fig. P15.127 and P15.128 15.128 Knowing that at the instant shown rod AB has zero angular accel
eration and an angular velocity of 15 rad/s counterclockwise, det_'  mine (a) the angular acceleration of member BD, (19) the acceleration
of point C. 15.129 Knowing that at the instant shown rod AB has a constant angti lar velocity of 6 rad/s clockwise, determine the acceleration99f
point D. . .. . ... ' '5 . . _ 1hr
225 mm 225 mm 15 130 Knowrng that at the instant shown rod AB has a constant angu velocity of 6 rad/s clockwise, determine (a) the angular accelera
Fig. P15.129 and P15.I30 tion of member BDE, (b) the acceleration of point E. PROBLEMS 16.1 A conveyor system is ﬁtted with vertical panels, and a BOO—mm
AB of mass 2.5 kg 15 lodged between two panels as shown. Kn
that the acceleration of the system is 1.5 m/s2 to the left, deter‘mjn.
(a) the force exerted on the rod at C, (b) the reaction at B. ' A conveyor system is ﬁtted with vertical panels, and a 300—mm_;r01
AB of mass 2. 5 kg 15 lodged between two panels as shown Ifl‘fth:
rod is to remain in the position shown, determine the maxim'
allowable acceleration of the system. A 6 ft board is placed in a truck with one end lesting againgt'ﬂ
block secured to the ﬂoor and the other leaning against a vertic'a]
partition. Determine the maximum allowable acceleration of1th
truck if the board is to remain in the position shown. Fig. PI6.'I and no.2 Fig. P16.3
A uniform rod BC weighing 8 lb is connected to a collar A by 1a
10 1n. cord AB. Neglecting the mass of the collar and cord, dezer
mine (a) the smallest constant acceleration aA for which the cord and the rod will lie in a straight line, (b) the corresponding tension
in the cord. Knowing that the coefﬁcient of static friction between the tires and
the road is 0.80 for the automobile shown, determine the maximum
possible acceleration on a level road, assuming (a) fourwheel drive
(b) rearwheel drive, (0) frontwheel drive. _' .. _ 60 in. 40 in. Fig. no.5 For the truck of Sample Prob. 16.1, determine the distance throug'
which the truck will skid if (a) the rearwheel brakes fail to operate
(12) the frontwheel brakes fail to operate. 7 A 20 kg cabinet IS mounted on casters that allow it to move freely
(,u — 0) on the ﬂoor. If a 100 N force is applied as shown, deter.
mine (a) the acceleration of the cabinet, (17) the range of values of
h for which the cabinet will not tip. ‘ Solve Prob. 16.7, assuming that the casters are locked and slide 011
Fig. P16.7 the rough ﬂoor (m, = 0.25). ' 1o4o Forces and Accelem'bns tion by three ropes as shown. Knowing that 0— — 30°, determm
immediately after rope CF has been cut, (a ) the acceleration of‘th ‘ ' 1042 ”One Mom" 0f Rigid Bodies: 16.14 A uniform rectangular plate has a mass of 5 kg and' 18 held 111'.qu
 plate, ([9) the tension in ropes AD and BE 300 m‘I—J Fig. P16.14 and P16J5 16.15 A uniform rectangular plate has a mass of 5 kg and is held in posi
tion by three ropes as shown Determine the largest value of 0 f0
which both ropes AD and BE remain taut immediately after rope
CF has been cut 16.16 A uniform circular plate of mass 3 kg is attached to two links AG
and BD of the same length. Knowing that the plate is released ' from rest in the position shown, determine (a ) the acceleration (if
Fig. P16.16 the plate, (17) the tension in each link. . 16.17 Three bars, each of weight 8 lb, are welded together and are p_ —
r—lsm'ﬂ connected to two links BE and CF. Neglecting the weight of the
links, determine the force' in each link immediately after the syst'e'gg 
is released from rest. 16.18 At the instant shown the angular velocity of links BE and CF is
6 rad/s counterclockwise and IS decreasing at the rate of 12 rad/st
Knowing that the length of each link IS 300 mm and neglecting the
weight of the links, determine (a) the force P, (b) the correspond:
ing force in each link. The mass of rod AD is 6 kg. Fig. P16.'l7 Fig. ”6.18 I 1046 Plane Motion of Rigid BodieS: 16.37 and 16.38 Two uniform disks and two cylinders are assemﬁle' Forces and Accelerations as indicated. Disk A weighs 20 lb and disk B weighs 12 lb. Knowm that the system is released from rest, determine the acceleraﬁm]
(a) of cylinder C, (b) of cylinder D. 16.37 Disks A and B are bolted together and the cylindém are attached to separate cords wrapped on the disks" 16.38 The cylinders are attached to a single cord that PaS§es over the disks. Assume that no slipping occurs bemé'e
the cord and the disks. .  Fig. P16.37 Fig. P1638 16.39 Disk A has a mass of 6 kg and an initial angular velocity of 360 rpm
clockwise; disk B has a mass of 3 kg and is initially at rest. The‘
disks are brought together by applying a horizontal force of
magnitude 20 N to the axle of disk A. Knowing that Mk = 0.15_
between the disks and neglecting bearing friction, determine}
(a) the angular acceleration of each disk, (1)) the ﬁnal angular velocil
ity of each disk. L Fig. P16.39 16.40 Solve Prob. 16.39, assuming that initially disk A is at rest and disk
B has an angular velocity of 360 rpm clockwise. 16.41 A belt of negligible mass passes between cylinders A and B and is
pulled to the right with a force P. Cylinders A and B weigh, respec
tively, 5 and 20 lb. The shaft of cylinder A is free to slide in a vertical.
slot and the coefficients of friction between the belt and each of the
cylinders are 1.1.; = 0.50 and #4: = 0.40. For P = 3.6 lb, determine
(a) whether slipping occurs between the belt and either cylinder, (19) the angular acceleration of each cylinder. 16.42 Solve Prob. 16.41 for P = 2.00 lb. Fig. no.4" 16.43 16.44 16.45 16.46 16.47 16.48 The 6lb disk A has a radius TA = 3 in. and an initial angular
velocity me = 375 rpm clockwise. The 15lb disk B has a radius
r3 = 5 in. and is at rest. A force P of magnitude 2.5 lb is then
applied to bring the disks into contact. Knowing that Mk = 0.25
between the disks and neglecting bearing friction, determine
(a) the angular acceleration of each disk, (19} the final angular
velocity of each disk. Solve Prob. 16.43, assuming that disk A is initially at rest and that
disk B has an angular velocity of 375 rpm clockwise. Disk B has an angular velocity (:00 when it is brought into contact
with disk A, which is at rest. Show that (a) the ﬁnal angular veloci
ties of the disks are independent of the coefﬁcient of friction #4:
between the disks as long as M 9’5 0, (b) the ﬁnal angular velocity
of disk B depends only upon (00 and the ratio of the masses mA
and m3 of the two disks. Show that the system of the effective forces for a rigid slab in plane
motion reduces to a single vector, and express the distance from
the mass center G of the slab to the line o_f action of this vector in
terms of the centroidal radius of gyration k of the slab, the magni
tude E of the acceleration of G, and the angular acceleration a. For a rigid slab in plane motion, show that the system of the effective
forces consists of vectors (Amiﬁ, —(Ami)w2r{ and (Ami)(a X 1'3
attached to the various particles P, of the slab, where a is the
acceleration of the mass center G of the slab, w is the angular
velocity of the slab, a is its angular acceleration, and 1'; denotes the
position vector of the particle Pi, relative to G. Further show, by
computing their sum and the sum of their moments about G, that
the effective forces reduce to a vector m5 attached at G and a
couple Ia. A uniform slender rod AB rests on a frictionless horizontal surface,
and a force P of magnitude 0.25 lb is applied at A in a direction
perpendicular to the rod. Knowing that the rod weighs 1.75 lb,
determine the acceleration of (a) point A, (19) point B. Fig. 916.48 16.49 (a) In Prob. 16.48, determine the point of the rod AB at which the force P should be applied if the acceleration of point B is to be
zero. (b) Knowing that P = 0.25 lb, determine the corresponding
acceleration of point A. Problems Fig. P16.43 and P16.45 [Amﬂéax 1",) Fig. P16.47 {mm)5 1047 H514.“ 16.56 16.57 16.59 16.60 16.55 A 3kg sprocket wheel has a centroidal radius of gyration of 70 mm and is suspended from a chain as shown. Determine the accelera
tion of points A and B of the chain, knowing that TA = 14 N and 80 mm Fig. P1655 Solve Prob. 16.55, assuming that TA = 14 N and TB = 12 N. and 16.58 A 15ft beam weighing 500 lb is lowered by means
of two cables unwinding from overhead cranes. As the beam
approaches the ground, the crane operators apply brakes to slow
the unwinding motion. Knowing that the deceleration of cable A
is 20 ft/s2 and the deceleration of cable B is 2 ft/sz, determine the tension in each cable. The steel roll shown has a mass of 1200 kg, a centriodal radius of
gyration of 150 mm, and is lifted by two cables looped around its
shaft. Knowing that for each cable TA = 3100 N and T3 = 3300 N,
determine (a) the angular acceleration of the roll, (I?) the accelera
tion of its mass center. The steel roll shown has a mass of 1200 kg, has a centriodal radius
of gyration of 150 mm, and is lifted by two cables looped around
its shaft. Knowing that at the instant shown the acceleration of the
roll is 150 mm/s2 downward and that for each cable TA = 3000 N,
determine (a) the corresponding tension TB, (19) the angular accel
eration of the roll. Fig. P1657 f" Fig. P1659 and P16.6o Problems 1049 1048 Plane Motion of Rigid Bodies: 16.50 and 16.51 A force P of magnitude 3 N is applied toast“
wrapped around the body indicated. Knowing that the body.196!
on a frictionless horizontal surface, determine the acceleraﬁo
(a) point A, (I?) point B.
16.50 A thin hoop of mass 2.4 kg.
16.51 A uniform disk of mass 2.4 kg. Forces and Accelerations no: Fig. P16.50 Fig. P16.51 and P16.52 16.52 A force P is applied to a tape wrapped around a uniform disk that
rests on a frictionless horizontal surface. Show that for each 360°
rotation of the disk the center of the disk will move a distance mg I 16.53 A 120kg satellite has a radius of gyration of 600 mm with respect
to the y axis and is symmetrical with respect to the zx plane. its
orientation is changed by ﬁring four small rockets A, B, C, and E, each of which produces a 16.20N thrust T directed as shown
Determine the angular acceleration of the satellite and the accel
eration of its mass center G (a) when all four rockets are ﬁred,‘ (1)) when all rockets except D are ﬁred. ' Fig. mass 16.54 A rectangular plate of mass 5 kg is suspended from four vertical—ll wires, and a force P of magnitude 6 N is applied to corner C as
shown. Immediately after P is applied, determine the acceleration a
of (a) the midpoint of edge BC, ([7) corner B. Fig. ”6.54 16.66 through 16.68 A thin plate of the shape indicated and of mass
m is suspended from two springs as shown. If spring 2 breaks,
determine the acceleration at that instant (a) of point A, (b) of
point B. 16.66 A circular plate of diameter 19. 16.67 A thin hoop of diameter 19. 16.68 A square plate of side 19. Fig. P16.66 Fig. P16.67
16.69 16.70 16.71 16.72 16.73 16.74 A bowler projects an 8in.diameter ball weighing 12 1b along an
alley with a forward velocity v0 of 15 ft/s and a backspin (no of
9 rad/s. Knowing that the coefﬁcient of kinetic friction between the
ball and the alley is 0.10, determine (a) the time t1 at which the
ball will start rolling without sliding, (b) the speed of the ball at
time t1, (0) the distance the ball will have traveled at time t1. Solve Prob. 16.69, assuming that the bowler projects the ball with
the same forward velocity but with a backspin of 18 rad/s. A sphere of radius 1* and mass m is projected along a rough hori
zontal surface with the initial velocities indicated. If the ﬁnal veloc
ity of the sphere is to be zero, express, in terms of 00, r, and ,uk,
((1) the required magnitude of (00, (b) the time t1 required for the
sphere to come to rest, (0) the distance the sphere will move before
coming to rest. Solve Prob. 16.71, assuming that the sphere is replaced by a uni  form thin hoop of radius 1" and mass m. A uniform sphere of radius r and mass m is placed with no initial
velocity on a belt that moves to the right with a constant velocity
v1. Denoting by ,th the coefficient of kinetic friction between the
sphere and the belt, determine (a) the time t1 at which the sphere
will start rolling without sliding, (b) the linear and angular veloci—
ties of the sphere at time t1. A sphere of radius r and mass m has a linear velocity v0 directed to
the left and no angular velocity as it is placed on a belt moving to
the right with a constant velocity v1. If after ﬁrst sliding on the belt the
sphere is to have no linear velocity relative to the ground as
it starts rolling on the belt without sliding, determine in terms of
01 and the coefﬁcient of kinetic friction ,uk between the sphere and
the belt (a) the required value of 00, (b) the time t1 at which the
sphere will start rolling on the belt, (a) the distance the sphere will
have moved relative to the ground at time t1. Problems 'I 05 'I Fig. P16.68 Fig. P16.69 Fig. P16.74 ...
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