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Unformatted text preview: PROBLEMS 19—1. The rigid body (slab) has a mass m and is rotating
with an angular velocity 1» about an axis passing through the
fixed point 0. Show that the momenta of all the particles
composing the body can be represented by a single vector
having a magnitude va and acting through point P,
called the center of percussion, which lies at a distance
rp/G = ké/rG/O from the mass center G. Here k6 is
the radius of gyration of the body, computed about an axis
perpendicular to the plane of motion and passing through G. Prob. 19—1 19—2. At a given instant, the body has a linear momentum
L = va and an angular momentum HG = [G to computed
about its mass center. Show that the angular momentum of
the body computed about the instantaneous center of zero
velocity [C can be expressed as Hm = 1mm, where [1C
represents the body‘s moment of inertia computed about
the instantaneous axis of zero velocity. As shown, the IC is
located at a distance rG/lc away from the mass center G. va \G) [ow Prob. 19—2 PROBLEMS 497 19—3. Show that if a slab is rotating about a fixed axis
perpendicular to the slab and passing through its mass
center G, the angular momentum is the same when
computed about any other point P on the slab. “Y
A? Prob. 19—3 *19—4. Gear A rotates along the inside of the circular gear
rack R. IfA has a weight of 4 lb and a radius of gyration of k3 = 0.5 ft, determine its angular momentum about point
C when mm = 30 rad/s and (3) (UR = 0, (b) wR = 20 rad/s. Prob. 19—4 498 CHAPTER 19 PLANAR KINETICS OF A RIGID BODY: IMPULSE AND MOMENTUM 19—5. Solve Prob. 17—55 using the principle of impulse and
momentum. 19—6. Solve Prob. 17—54 using the principle of impulse and
momentum. 19—7. Solve Prob. 17—69 using the principle of impulse and
momentum. *19—8. Solve Prob. 17—80 using the principle of impulse
and momentum. 19—9. Solve Prob. 17—73 using the principle of impulse and
momentum. 19—10. A ﬂywheel has a mass of 60 kg and a radius of
gyration of kg 2 150 mm about an axis of rotation passing
through its mass center. If a motor supplies a clockwise
torque having a magnitude of M = (52‘) N  m, where t is in
seconds, determine the ﬂywheel’s angular velocity in
t = 3 5. Initially the flywheel is rotating clockwise at
w] = 2 rad/s. 19—11. A wire of negligible mass is wrapped around the
outer surface of the 2kg disk. If the disk is released from
rest, determine its angular velocity in 3 s. Prob. 19—11 *19—12. The spool has a mass of 30 kg and a radius of
gyration k0 = 0.25 m. Block A has a mass of 25 kg, and
block B has a mass of 10 kg. If they are released from rest,
determine the time required for block A to attain a speed of
2 m/s. Neglect the mass of the ropes. Prob. 19—12 19—13. The man pulls the rope off the reel with a constant
force of 8 lb in the direction shown. If the reel has a weight
of 250 lb and radius of gyration k6 = 0.8 ft about the
trunnion (pin) at A, determine the angular velocity of the
reel in 3 5 starting from rest. Neglect friction and the weight
of rope that is removed. Prob. 19—13 19—14. Angular motion is transmitted from a driver
wheel A to the driven wheel B by friction between the
wheels at C. If A always rotates at a constant rate of
16 rad/s, and the coefficient of kinetic friction between the
wheels is Mk = 0.2, determine the time required for B to
reach a constant angular velocity once the wheels make
contact with a normal force of 50 N. What is the final
angular velocity of wheel B? Wheel B has a mass of 90 kg
and a radius of gyration about its axis of rotation of
kc = 120 mm. Prob. 19—14 19—15. The 4kg slender rod rests on a smooth floor. If it is
kicked so as to receive a horizontal impulse I = 8 N  s at
point A as shown, determine its angular velocity and the
speed of its mass center. PROBLEMS 499 *19—16. A cord of negligible mass is wrapped around the
outer surface of the 50lb cylinder and its end is subjected to
a constant horizontal force of P = 2 lb. If the cylinder rolls
without slipping at A, determine its angular velocity in 4 5
starting from rest. Neglect the thickness of the cord. P=21b A.—..N~—_...._.__~ Prob. 19—16 19—17. The drum has a mass of 70 kg, a radius of 300 mm,
and radius of gyration k0 = 125 mm. If the coefficients of
static and kinetic friction at A are ,us = 0.4 and ,uk = 0.3,
respectively, determine the drum’s angular velocity 2 s after
it is released from rest.Take 0 = 30°. Prob. 19—15 Prob. 19—17 500 CHAPTER 19 19—18. The double pulley consists of two wheels which are
attached to one another and turn at the same rate. The
pulley has a mass of 15 kg and a radius of gyration
k0 = 110mm. If the block at A has a mass of 40 kg,
determine the speed of the block in 3 s after a constant
force F = 2 kN is applied to the rope wrapped around the
inner hub of the pulley. The block is originally at rest.
Neglect the mass of the rope. Prob. 1918 19—19. The spool has a weight of 30 lb and a radius of
gyration k0 : 0.45 ft. A cord is wrapped around its inner
hub and the end subjected to a horizontal force P = 5 lb.
Determine the spool’s angular velocity in 4 3 starting from
rest. Assume the spool rolls without slipping. Prob. 19—19 PLANAR KINETICS OF A RIGID BODY: IMPULSE AND MOMENTUM *19——20. The two gears A and B have weights and radii of
gyration of WA = 15 lb, k4 = 0.5 ft and W8 = 101b,
k3 = 0.35 ft, respectively. If a motor transmits a couple
moment to gear .8 of M = 2(1 — 605’) lb ~ fti where {is in
seconds, determine the angular velocity of gear A in I = 5 5,
starting from rest. Prob. 19—20 1921. Spool B is at rest and spool A is rotating at
6rad/s when the slack in the cord connecting them is
taken up. Determine the angular velocity of each spool
immediately after the cord is jerked tight by the spinning of
spool A. The weights and radii of gyration of A and B
are WA = 301b,kA = 0.8 ft and W3 = 151b,k3 = 0.6 ft,
respectively. Prob. 19—21 19—22. A 4kg disk A is mounted on arm BC, which has a
negligible mass. If a torque of M = (5e0'5’) N  m, where tis
in seconds, is applied to the arm at C, determine the angular
velocity of BC in 2 5 starting from rest. Solve the problem
assuming that (a) the disk is set in a smooth bearing at B so
that it rotates with curvilinear translation, (b) the disk is
fixed to the shaft BC, and (c) the disk is given an initial
freely spinning angular velocity of (up = {—80k} rad/s
prior to application of the torque. Prob. 19—22 19—23. The inner hub of the wheel rests on the horizontal
track. If it does not slip at A, determine the speed of the
101b block in 2 s after the block is released from rest.
The wheel has a weight of 30 lb and a radius of gyration
kc = 1.30 ft. Neglect the mass of the pulley and cord. Prob. 19—23 PROBLEMS 5 01 *19—24. If the hoop has a weight W and radius r and is
thrown onto a rough surface with a velocity VG parallel to
the surface, determine the amount of backspin, (no, it must
be given so that it stops spinning at the same instant that its
forward velocity is zero. It is not necessary to know the
coefficient of kinetic friction at A for the calculation. Prob. 19—24 19—25. The 10lb rectangular plate is at rest on a smooth
horizontal floor. If it is given the horizontal impulses
shown, determine its angular velocity and the velocity of the
mass center. y Slbs Prob. 19—25 502 CHAPTER 19 PLANAR KINETICS OF A RIGID BODY: IMPULSE AND MOMENTUM 1926. The ball of mass m and radius r rolls along an *19—28. The slender rod has a mass m and is suspended
inclined plane for which the coefficient of static friction is n. at its end A by a cord. If the rod receives a horizontal
If the ball is released from rest, determine the maximum blow giving it an impulse I at its bottom B, determine the
angle 0 for the incline so that it rolls without slipping at A. location y of the point P about which the rod appears to rotate during the impact. Prob. 19—26
Prob. 19—28
19—27. The spool has a weight of 75 lb and a radius of 19—29. Athin rod havingamass of 4 kg is balanced vertically
gyration k0 = 1.20 ft. If the block B weighs 60 lb, and a as shown. Determine the height h at which it can be struck
force F = 25 lb is applied to the cord, determine the speed with a horizontal force F and not slip on the floor. This requires
of the block in 5 5 starting from rest. Neglect the mass of that the frictional force atA be essentially zero.
the cord. Prob. 19—27 Prob. 19—29 19—30. The square plate has a mass m and is suspended at
its corner A by a cord. If it receives a horizontal impulse I at
corner B, determine the location y of the point P about
which the plate appears to rotate during the impact. Prob. 19—30 19—31. Determine the height h of the bumper of the pool
table, so that when the pool ball of mass m strikes it, no
frictional force will be developed between the ball and the
table at A. Assume the bumper exerts only a horizontal
force on the ball. Prob. 19—31 PROBLEMS 503 *19—32. The double pulley consists of two wheels which
are attached to one another and turn at the same rate. The
pulley has a mass of 15 kg and a radius of gyration of
k0 = 110 mm. If the block at A has a mass of 40 kg and the
container at B has a mass of 85 kg, including its contents,
determine the speed of the container when t = 3 s after it is
released from rest. Prob. 19—32 19—33. The crate has a mass m6. Determine the constant
speed 120 it acquires as it moves down the conveyor. The
rollers each have a radius of r, mass m, and are spaced d
apart. Note that friction causes each roller to rotate when
the crate comes in contact with it. Prob. 19—33 512 CHAPTER 19 PLANAR KINETICS OF A RIGID BODY: IMPULSE AND MOMENTUM PROBLEMS 19—34. Two wheels A and B have masses mA and m3, and
radii of gyration about their central vertical axes of k A and
k3, respectively. If they are freely rotating in the same
direction at 00A and tag about the same vertical axis,
determine their common angular velocity after they are
brought into contact and slipping between them stops. 1935. The Hubble Space Telescope is powered by
two solar panels as shown. The body of the telescope has a
mass of 11 Mg and radii of gyration k x = 1.64 m and
k\. = 3.85 m, whereas the solar panels can be considered as
thin plates, each having a mass of 54 kg. Due to an internal
drive, the panels are given an angular velocity of {0.6j} rad/s,
measured relative to the telescope. Determine the angular
velocity of the telescope due to the rotation of the panels.
Prior to rotating the panels, the telescope was originally
traveling at VG = {—400i + 250j + 175k} m/s. Neglect its
orbital rotation. Prob. 1935 *19—36. The platform swing consists of a ZOOlb ﬂat plate
suspended by four rods of negligible weight. When the
swing is at rest, the 150lb man jumps off the platform when
his center of gravity G is 10 ft from the pin at A. This is done
with a horizontal velocity of 5 ft/s, measured relative to the
swing at the level of G. Determine the angular velocity he
imparts to the swing just after jumping off. Prob. 19—36 19—37. Each of the two slender rods and the disk have the
same mass m. Also, the length of each rod is equal to the
diameter d of the disk. If the assembly is rotating with an
angular velocity 001 when the rods are directed outward,
determine the angular velocity of the assembly if
by internal means the rods are brought to an upright
vertical position. Prob. 19—37 19—38. The rod has a length L and mass m. A smooth
collar having a negligible size and onefourth the mass of
the rod is placed on the rod at its midpoint. If the rod is
freely rotating at u) about its end and the collar is released,
determine the rods angular velocity just before the collar
flies off the rod. Also. what is the speed of the collar as it
leaves the rod? Prob. 19—38 19—39. A man has a moment of inertia I: about the z axis.
He is originally at rest and standing on a small platform
which can turn freely. If he is handed a wheel which is
rotating at w and has a moment of inertia I about its
spinning axis, determine his angular velocity if (a) he holds
the wheel upright as shown, (b) turns the wheel out,
0 = 90°, and (c) turns the wheel downward, 0 = 180°. PROBLEMS 51 3 *19—40. The space satellite has a mass of 125 kg and a
moment of inertia IZ = 0.940 kgmz, excluding the four
solar panels A, B, C, and D. Each solar panel has a mass of
20 kg and can be approximated as a thin plate. If the
satellite is originally spinning about the z axis at a constant
rate a)z = 0.5 rad/s when 6 = 90°, determine the rate of
spin if all the panels are raised and reach the upward
position, 6 = 0°, at the same instant Prob. 19—40 19—41. The 2—kg rod ACB supports the two 4—kg disks at
its ends. If both disks are given a clockwise angular velocity
(cu/4)] = (wB)1 = 5 rad/s while the rod is held stationary
and then released, determine the angular velocity of the rod
after both disks have stopped spinning relative to the rod
due to frictional resistance at the pins A and 8. Motion is in
the horizontal plane. Neglect friction at pin C. Prob. 19—39 Prob. 1941 514 CHAPTER 19 PLANAR KINETICS OF A RIGID BODY: IMPULSE AND MOMENTUM 19—42. Disk A has a weight of 20 lb. An inextensible cable
is attached to the 10lb weight and wrapped around the
disk. The weight is dropped 2 ft before the slack is taken up.
If the impact is perfectly elastic, i.e., e = 1, determine the
angular velocity of the disk just after impact. Prob. 19—42 19—43. A thin disk of mass m has an angular velocity wl
while rotating on a smooth surface. Determine its new
angular velocity just after the hook at its edge strikes the
peg P and the disk starts to rotate about P without
rebounding. Prob. 19—43 *19—44. The pendulum consists of a 5lb slender rod AB
and a 10lb wooden block. A projectile weighing 0.2 lb is
fired into the center of the block with a velocity of 1000 ft/s.
If the pendulum is initially at rest, and the projectile embeds
itself into the block, determine the angular velocity of the
pendulum just after the impact. 2) =1000ft/s ' Prob. 19—44 19—45. The pendulum consists of a slender 2kg rod AB
and 5kg disk. It is released from rest without rotating.
When it falls 0.3 m, the end A strikes the hook S, which
provides a permanent connection. Determine the angular
velocity of the pendulum after it has rotated 90°. Treat the
pendulum’s weight during impact as a nonimpulsive force. Prob. 19—45 PROBLEMS 51 5 19—46. A horizontal circular platform has a weight of *19~48. Two children A and B, each having a mass of 30 300 lb and a radius of gyration kz = 8 ft about the z axis kg, sit at the edge of the merrygoround which is rotating at
passing through its center 0. The platform is free to rotate w = 2 rad/s. Excluding the children, the merry—goround
about the z axis and is initially at rest. A man having a has a mass of 180 kg and a radius of gyration kZ = 0.6 m.
weight of 150 lb begins to run along the edge in a circular Determine the angular velocity of the merrygoround if A
path of radius 10ft. If he has a speed of 4 ft/s and maintains jumps off horizontally in the —n direction with a speed of
this speed relative to the platform, determine the angular 2 m/s, measured with respect to the merrygo—round. What
velocity of the platform. Neglect friction. is the merrygo—round’s angular velocity if B then jumps off horizontally in the +1? direction with a speed of 2m/s,
measured with respect to the merrygoround? Neglect
friction and the size of each child. Prob. 19—46
a) = 2 rad /s Prob. 19—48 1947. The square plate has a weight W and is rotating on 19—49. A 7g bullet having a velocity of 800 m/s is fired the smooth surface with a constant angular velocity wo. into the edge of the Skg disk as shown. Determine the
Determine the new angular velocity of the plate just after angular velocity of the disk just after the bullet becomes
its corner strikes the peg P and the plate starts to rotate embedded in it. Also, calculate how far 9 the disk will swing
about P without rebounding. until it stops. The disk is originally at rest. Prob. 19—47 Prob. 19—49 516 CHAPTER 19 19—50. The two disks each weigh 10 lb. If they are released
from rest when 6 = 30°, determine 0 after they collide and
rebound from each other. The coefficient of restitution is
e = 0.75. When 0 = 0°, the disks hang so that they just
touch one another. PLANAR KINETICS OF A RIGID BODY: IMPULSE AND MOMENTUM *19—52. The pendulum consists of a 10lb solid ball and
4—lb rod. If it is released from rest when 61 = 0°, determine
the angle (92 after the ball strikes the wall, rebounds, and the
pendulum swings up to the point of momentary rest. Take 6 = 0.6. Prob. 1950 19—51. The 15lb rod AB is released from rest in the
vertical position. If the coefficient of restitution between
the floor and the cushion at B is e = 0.7, determine how
high the end of the rod rebounds after impact with
the floor. 2ft Prob. 19—51 Prob. 19—52 1953. The plank has a weight of 30 lb, center of gravity at
G, and it rests on the two sawhorses at A and B. If the end D
is raised 2 ft above the top of the sawhorses and is released
from rest, determine how high end C will rise from the top
of the sawhorses after the plank falls so that it rotates
clockwise about A, strikes and pivots on the sawhorses at B,
and rotates clockwise off the sawhorse at A. i
l
% 3 ft 4.1.5 ft++l.5 MW 3 ft—J Prob. 19—53 19—54. Tests of impact on the fixed crash dummy are
conducted using the SODlb ram that is released from rest at
6 = 30°, and allowed to fall and strike the dummy at
6 = 90°. If the coefficient of restitution between the dummy
and the ram is e = 0.4‘ determine the angle 9 to which the
ram will rebound before momentarily coming to rest. Prob. 19—54 19—55. The solid ball of mass m is dropped with a velocity
v1 onto the edge of the rough step. If it rebounds
horizontally off the step with a velocity v2, determine the
angle 6 at which contact occurs. Assume no slipping when
the ball strikes the step. The coefficient of restitution is e. Prob. 19—55 PROBLEMS 51 7 *19—56. A solid ball with a mass m is thrown on the
ground such that at the instant of contact it has an angular
velocity ml and velocity components (v(;)x1 and 06)“ as
shown. If the ground is rough so no slipping occurs.
determine the components of the velocity of its mass center
just after impact. The coefficient of restitution is e. Prob. 19—56 ...
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This note was uploaded on 04/16/2008 for the course CE 325 taught by Professor Docwong during the Spring '08 term at USC.
 Spring '08
 DocWong

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