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Unformatted text preview: 16.88 16.89 16.90 16.91 16.92 16.93 An 8lb slender rod AB and a 5—lb slender rod BC are connected
by a pin at B and by the cord AC. The assembly can rotate in a
vertical plane under the combined effect of gravity and a couple
M applied to rod BC. Knowing that in the position shown the
angular velocity of the assembly is zero and the tension in cord AC
is equal to 6 lb, determine (a) the angular acceleration of the
assembly, (19) the magnitude of the couple M. 12 in.——>“— 12in. Fig. P16.88 Two uniform rods, ABC of mass 3 kg and DCE of mass 4 kg,
are connected by a pin at C and by two cords BD and BE. The
T—shaped assembly rotates in a vertical plane under the combined
effect of gravity and of a couple M which is applied to rod ABC.
Knowing that at the instant shown the tension is 8 N in cord BD,
determine (a) the angular acceleration of the assembly, {19] the
couple M. A 1.5—kg slender rod is welded to a 5kg uniform disk as shown.
The assembly swings freely about C in a vertical plane. Knowing
that in the position shown the assembly has an angular velocity of
10 rad/s clockwise, determine (a) the angular acceleration of the
assembly, (b) the components of the reaction at C. Fig. "6.90 A 5kg uniform disk is attached to the 3kg uniform rod BC by
means of a frictionless pin AB. An elastic cord is wound around
the edge of the disk and is attached to a ring at E. Both ring E
and rod BC can rotate freely about the vertical shaft. Knowing that
the system is released from rest when the tension in the elastic
cord is 15 N, determine (a) the angular acceleration of the disk,
(1)) the acceleration of the center of the disk. Derive the equation EMC = Ica for the rolling disk of Fig. 16.17,
where EMC represents the sum of the moments of the external forces about the instantaneous center C, and Ic is the moment of
inertia of the disk about C. Show that in the case of an unbalanced disk, the equation derived
in Prob. 16.92 is valid only when the mass center G, the geometric
center 0, and the instantaneous center C happen to lie in a straight
line. Fig. P16.89 Problems 1 065 Z Fig. P1691 Plane Motion of Rigid Bodies: Forces and Accelerations 16.95
Fig. P16.94
16.96
cc “INK. " '. 'is I 16.97
[3: 0° Fig. 916.97
16.98 Fig. P1699 and P16.103 Fig. P16.100 and P16.104 16.94 A wheel of radius r and centroidal radius of gyration I: is released fmm rest on the incline and rolls without sliding. Derive an exp_ression for
the acceleration of the center of the wheel in terms of r, k, [3,311ng A flywheel is rigidly attached to a shaft of 1.5in. radius that can
roll along parallel rails as shown. When released from rest, the
system rolls 16 ft in 40 5. Determine the centroidal radius of gyra~
tion of the system. 15% Fig. 916.95 and P1696
A ﬂywheel of centroidal radius of gyration k is rigidly attached to
a shaft that can roll along parallel rails. Denoting by ,u, the coefﬁ.
cient of static friction between the shaft and the rails, derive an
expression for the largest angle of inclination B for which no
slipping will occur. A homogeneous sphere S, a uniform cylinder C, and a thin pipe P
are in contact when they are released from rest on the incline
shown. Knowing that all three objects roll without slipping, deter
mine, after 4 s of motion, the clear distance between (a) the pipe
and the cylinder, (19) the cylinder and the sphere. through 16.101 A drum of 4in. radius is attached to a disk of
8in. radius. The disk and drum have a combined weight of 10 lb
and a combined radius of gyration of 6 in. A cord is attached as
shown and pulled with a force P of magnitude 5 lb. Knowing that
the coefﬁcients of static and kinetic friction are ,u, = 0.25 and M = 0.20, respectively, determine (a) whether or not the disk slides,
(17) the angular acceleration of the disk and the acceleration of G. 16.102 through 16.105 A drum of 60mm radius is attached to a disk
of 120mm radius. The disk and drum have a total mass of 6 kg and
a combined radius of gyration of 90 mm. A cord is attached as
shown and pulled with a force P of magnitude 20 N. Knowing that
the disk rolls without sliding, determine (a) the angular acceleraﬁon
of the disk and the acceleration of G, (b) the minimum value of the
coefﬁcient of static friction compatible with this motion. Fig. P16.101 and P16.105 Plane Motion of Rigid BOdieSi 16.113 A small clamp of mass m3 is attached at B to a hoop of mass ma.
Fm“ °nd Amaleml'ms The system is released from rest when 0 = 90a and rolls without  n
sliding. Knowing that mh = 3mg, determine (a) the angular accel §\ eration of the hoop, (b) the horizontal and vertical components of
9
A (>3 the acceleration of B. 16.114 A small clamp of mass m3 is attached at B to a hoop of mass mh.
\ Knowing that the system is released from rest and rolls Without
\ j." sliding, derive an expression for the angular acceleration of the hoop in terms of mg, m;,, r, and 0. 16.115 The center of gravity G of a 1.5kg unbalanced tracking wheel is Fig. p16_"3 and “6.,” located at a distance r = 18 mm from its geometric center B. The radius of the wheel is R = 60 mm and its centroidal radius of r: 18 mm gyration is 44 mm. At the instant Shown the center B of the Wheel ' has a velocity of 0.35 m/s and an acceleration of 1.2 m/52, both directed to the left. Knowing that the wheel rolls without sliding and neglecting the mass of the driving yoke AB, determine the
horizontal force P applied to the yoke. 16.116 A 2kg bar is attached to a 5kg uniform cylinder by a square pin, 1
P, as shown. Knowing that r = 0.4 m, h = 0.2 m, 0 = 20°, L =
0.5 m and a) = 2 rad/s at the instant shown, determine the reac
tions at P at this instant assuming that the cylinder rolls without
sliding down the incline. 16.117 The ends of the lO—kg uniform rod AB are attached to collars of I
negligible mass that slide without friction along fixed rods. If the '
rod is released from rest when 0 = 25°, determine immediately
after release {a} the angular acceleration of the rod, (19} the reaction
at A, (b) the reaction at B. Fig. P16.117 and P16.118 30in 16.118 The ends of the lO—kg uniform rod AB are attached to collars of ' ’ negligible mass that slide Without friction along ﬁxed rods. A verti
cal force P is applied to collar B when 9 = 25°, causing the collar
to start from rest with an upward acceleration of 12 m/32. Deter—
mine (a) the force P, (b) the reaction at A. l . //  0 of negligible weight that roll along without friction in the slots
shown. If the rod is released from rest in the position shown, deter— l
L—— mine immediately after release (a) the angular acceleration of the
Fig. P16.‘Il9 rod, {bi the reaction at B. 3 16.119 The motion of the 8lb uniform rod AB is guided by small wheels Plane Motion of Rigid Bodies: 16.125 The 250mm uniform rod BD, of mass 5 kg, is connected as shown 16.
Fm“ and Accelem'wns to disk A and to a collar of negligible mass, which may slide freely
W along a vertical rod. Knowing that disk A rotates countercloclwvjse _ at a constant rate of 500 rpm, determine the reactions at D when 0 = 0. 16.126 Solve Prob. 16.125 when 9 — 90°. Q) 16
16.127 The 15in. uniform rod BD weighs 8 lb and is connected as shown l to crank AB and to a collar D of negligible weight, which can slide freely along a horizontal rod. Knowing that crank AB rotates coun. terclockwise at the constant rate of 300 rpm, determine the reac 16.
tion at D when 0 = 0. 16.128 Solve Prob. 16.127 when 0 = 90°. 16.129 The 3kg uniform rod AB is connected to crank BD and to a collar 16.
of negligible weight, which can slide freely along rod EF. Knowing
that in the position shown crank BD rotates with an angular veloc—
ity of 15 rad/s and an angular acceleration of 60 rad/$2, both clock
wise, determine the reaction at A. 16.
‘ 500 mm _—‘


' D o
80 mm
_ _ '9; —L
161
16: Fig. P16.129 16.130 In Prob. 16.129, determine the reaction at A, knowing that in
the position shown crank BD rotates with an angular velocity of
15 rad/s clockwise and an angular acceleration of 60 rad/s2
counterclockwise. Fig. mam 16.131 A driver starts his car with the door on the passenger’s side wide
open {0 = 0). The 80lb door has a centroidal radius of gyration
k = 12.5 and its mass center is located at a distance r = 22 in.
from its vertical axis of rotation. Knowing that the driver maintains a constant acceleration of 6 ft/s2, determine the angular velocity of
the door as it slams shut (0 = 90°). 16.132 For the car of Prob. 16.131, determine the smallest constant accel
eration that the driver can maintain if the door is to close and latch,
knowing that as the door hits the frame its angular velocity must be
at least 2 rad/s for the latching mechanism to operate. f... AL.
3 16.133 Two 8lb uniform bars are connected to form the linkage shown. N 1 . . . . .
15 in. 15in. eg ectmg the effect of friction, determine the reaction at D immediately after the linkage is released from rest in the position
Fig. P16.'l33 shown. Plane Motion of Rigid Bodies: Energy and Momentum Methods 17.8 I 7.9 Fig. P17.7 and P'I7.8 ‘ A B _ ll 80 mm maﬁa QSJU‘L 80 mm 'I7.'0 17." Fig. P'I7.9 17.12 Fig. 917.12 17.7 Disk A is of constant thickness and is at rest when it is placed in contact with belt BC, which moves with a constant velocity v‘
Denoting by [.Lk the coefficient of kinetic friction between the disk
and the belt, derive an expression for the number of revolutions
executed by the disk before it attains a constant angular velocity. Disk A, of weight 10 lb and radius r — 6 in., is at rest when it is
placed in contact with belt BC, which moves to the right with a
constant speed 0 = 40 ft/s. Knowing that m. = 0.20 between the
disk and the belt, determine the number of revolutions executed
by the disk before it attains a constant angular velocity. Each of the gears A and B has a mass of 2.4 kg and a radius of gym.
tion of 60 mm, while gear C has a mass of 12 kg and a radius of
gyration of 150 mm. A couple M of constant magnitude 10 N  m is
applied to gear C. Determine (a) the number of revolutions of gear
C required for its angular velocity to increase from 100 to 450 rpm)
(19) the corresponding tangential force acting on gear A. Solve Prob. 17.9, assuming that the lON ' m couple is applied to
gear B. The double pulley shown weighs 30 lb and has a centroidal radius
of gyration of 6.5 in. Cylinder A and block B are attached to cords
that are wrapped on the pulleys as shown. The coefficient of
kinetic friction between block B and the surface is 0.25. Knowing
that the system is released from rest in the position shown, deter
mine (a) the velocity of cylinder A as it strikes the ground, (19) the
total distance that block B moves before coming to rest. Fig. PI7.'I‘I The 8—in.—radius brake drum is attached to a larger flywheel that
is not shown. The total mass moment of inertia of the ﬂywheel and
drum is 14 lb  ft  s2 and the coefficient of kinetic friction between
the drum and the brake shoe is 0.35. Knowing that the initial
angular velocity of the flywheel is 360 rpm counterclockwise,
determine the vertical force P that must be applied to the pedal
C if the system is to stop in 100 revolutions. 17.13 Solve Prob. 17.12, assuming that the initial angular velocity of the “05'9"” ﬂywheel is 360 rpm clockwise. 17.14 The gear train shown consists of four gears of the same thickness and of the same material; two gears are of radius r, and the other two are of radius 111'. The system is at rest when the couple M0 is applied to shaft C. Denoting by Io the moment of inertia of a gear of radius r, determine the angular velocity of shaft A if the couple
M0 is applied for one revolution of shaft C.
l Fig. 1917.14 17.15 The three friction disks shown are made of the same material and l/A ' _ __ _ have the same thickness. It is known that disk A weighs 12 lb /’ //3 .—._‘ i
and that the radii of the disks are 134 = 8 in., n; = 6 in., and / rA f ra>\f/6 r9 _. l
rc = 4 in. The system is at rest when a couple M0 of constant magni I: Mn f ﬂ '[ I " '
tude 60 lb ~ in. is applied to disk A. Assuming that no slipping occurs "1, '\.\ I
between disks, determine the number of revolutions required for \_ x" \m // ’
disk A to reach an angular velocity of 150 rpm. NEH—J _ Fig. P17.15
17.16 and 17.17 A slender 4kg rod can rotate in a vertical plane
about a pivot at B. A spring of constant k = 400 N/m and of
unstretched length 150 mm is attached to the rod as shown. Know
ing that the rod is released from rest in the position shown, deter— ]
mine its angular velocity after it has rotated through 90". 600 mm Fig. P17.16 Fig. P17.17 Plane Motion of Rigid Bodies: Energy and 17.18 A slender rod of length l and weight W is pivoted at one end as
M°mem°m Melh°ds shown. It is released from rest in a horizontal position and swin
freely. ((1) Determine the angular velocity of the rod as it passes through a vertical position and determine the corresponding man.
ALT: __ tion at the pivot. (b) Solve part a for W = 1.8 lb andl = 3 ft. B
l 17.19 A slender rod of length l is pivoted about a point C located at a Fig. "7.18 distance 12 from its center C. It is released from rest in a horizontal
i position and swings freely. Determine (a) the distance In for which
I the angular velocity of the rod as it passes through a vertical posi_
tion is maximum, (19) the corresponding values of its angular veloc ity and of the reaction at C. Fig. 1917.19 17.20 A l60lb gymnast is executing a series of fullcircle swings on the
horizontal bar. In the position shown he has a small and negligible
clockwise angular velocity and will maintain his body straight and
rigid as he swings downward. Assuming that during the swing the
centroidal radius of gyration of his body is 1.5 ft, determine his
angular velocity and the force exerted on his hands after he has
rotated through (a) 90°, ([9) 180°. 17.21 Two identical slender rods AB and BC are welded together to form
an L—shaped assembly. The assembly is pressed against a spring at
D and released from the position shown. Knowing that the maxi
mum angle of rotation of the assembly in its subsequent motion is
90° counterclockwise, determine the magnitude of the angular
velocity of the assembly as it passes through the position where
rod AB forms an angle of 30° with the horizontal. 17.22 A collar with a mass of 1 kg is rigidly attached at a distance d =
300 mm from the end of a uniform slender rod AB. The rod has a
mass of 3 kg and is of length L = 600 mm. Knowing that the rod
is released from rest in the position shown, determine the angular
velocity of the rod after it has rotated through 90°. 17.23 A collar with a mass of 1 kg is rigidly attached to a slender rod AB
of mass 3 kg and length L = 600 mm. The rod is released from rest
in the position shown. Determine the distance d for which the angu
lar velocity of the rod is maximum after it has rotated through 90°. 0.4 m 17.24 A 20kg uniform cylindrical roller, initially at rest, is acted upon
by a 90N force as shown. Knowing that the body rolls without
slipping, determine (a) the velocity of its center C after it has
moved 1.5 In, (19) the friction force required to prevent slipping. Fig. 1917.21 ‘\ L G ___ 250mm
[v k—d—:l : 90 N
C . I.
L _ ' 3‘21! /
A B Fig. P17.22 and P17.23 Fig. P1124 17.41 17.42 17.43 17.44 17.45 17.46 The motion of a slender rod of length R is guided by pins at A and
B which slide freely in slots cut in a vertical plate as shown. If end
B is moved slightly to the left and then released, determine the
angular velocity of the rod and the velocity of its mass center
(a) at the instant when the velocity of end B is zero, lb] as end B passes through point D. Fig. P1141 Two uniform rods, each of mass m and length L, are connected to
form the linkage shown. End D of rod BD can slide freely in the
horizontal slot, while end A of rod AB is supported by a pin and
bracket. If end D is moved slightly to the left and then released,
determine its velocity (a) when it is directly below A, i ) when rod
AB is vertical. The uniform rods AB and BC weigh 2.4 lb and 4 lb, respectively,
and the small wheel at C is of negligible weight. If the wheel is
moved slightly to the right and then released, determine the veloc
ity of pin B after rod AB has rotated through 90”. Fig. P1143 and P17.44 The uniform rods AB and BC weigh 2.4 lb and 4 lb, respectively,
and the small wheel at C is of negligible weight. Knowing that in the
position shown the velocity of wheel C is 6 ft/s to the right, determine
the velocity of pin B after rod AB has rotated through 90”. The 4kg rod AB is attached to a collar of negligible mass at A and
to a flywheel at B. The flywheel has a mass of 16 kg and a radius
of gyration of 180 mm. Knowing that in the position shown the
angular velocity of the ﬂywheel is 60 rpm clockwise, determine
the velocity of the flywheel when point B is directly below C. If in Prob. 17.45 the angular velocity of the ﬂywheel is to be the
same in the position shown and when point B is directly above C,
determine the required value of its angular velocity in the position shown. Problems .0552.
T_ l E1::ij D Fig. P17.42 Fig. P17.45 and P17.46 1101 ...
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