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PROBLEM 17.76 (Continued)
PROBLEM 17.77
A sphere of radius r and mass m is projected along a rough horizontal surface
with the initial velocities shown. If the final velocity of the sphere is to be zero,
express (a) the required magnitude of
0
in terms of v
0
and r, (b) the time
required for the sphere to come to rest in terms of v
0
and coefficient of kinetic
friction
.
k
SOLUTION
Moment of inertia. Solid sphere.
2
2
5
Imr
PROBLEM 17.78
A bowler projects an 8.5-in.-diameter ball weighing 16 lb along an alley with a forward
velocity v0 of 25 ft/s and a backspin
0 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.
SOLUTION
55
Use the principle of impulse and momentum.
1
S
y
st. Momenta 12
S
y
st. Ext. Im
p
.
2
S
y
st. Momenta
PROBLEM 17.78 (Continued)
PROBLEM 17.79
A semicircular panel of radius r is attached with hinges to a circular plate of radius
r and initially held in the vertical position as shown. The plate and the panel are
made of the same material and have the same thickness. Knowing that the entire
assembly is rotating freely with an initial angular velocity 0,
determine the
angular velocity of the assembly after the panel has been released and comes to rest
against the plate.
SOLUTION
Let m be the mass of the plate. Then, the mass of the panel is 1.
20
6
PROBLEM 17.80
A satellite has a total weight (on Earth) of 250 lbs, and each of the solar panels
weighs 15 lbs. The body of the satellite has mass moment of inertia about the z-
axis of 6 slug-ft2, and the panels can be modeled as flat plates. The satellite
spins with a rate of 10 rpm about the z-axis when the solar panels are
positioned in the xy plane. Determine the spin rate about z after a motor on the
satellite has rotated both panels to be positioned in the yz plane (as shown in the
figure).
SOLUTION
250 slugs, 6 slug ft
3
Conservation of Momentum about the z-axis:
,1 1 , 2 2
,1
21
,2
OO
OO
O
O
II
I
I
Mass Moment of Inertia about the z-axis with panels in the first position:
,1 ,1
2
,1
2
22
2
2
62
1
62 4 10 7.5
12
67.413 slug ft
Os p
ppp
pp
III
Imd
mm
Mass Moment of Inertia about the z-axis with panels in the second position:
PROBLEM 17.81
Two 10-lb disks and a small motor are mounted on a 15-lb rectangular
platform which is free to rotate about a central vertical spindle. The
normal operating speed of the motor is 180 rpm. If the motor is started
when the system is at rest, determine the angular velocity of all elements
of the system after the motor has attained its normal operating speed.
Neglect the mass of the motor and of the belt.
1
12
2
PROBLEM 17.81 (Continued)
Moments about O:
20.9 rpm
PROBLEM 17.82
A 3-kg rod of length 800 mm can slide freely in the 240-mm cylinder
DE, which in turn can rotate freely in a horizontal plane. In the
position shown the assembly is rotating with an angular velocity of
magnitude 40 rad/s
and end B of the rod is moving toward the
cylinder at a speed of 75 mm/s relative to the cylinder. Knowing that
the centroidal mass moment of inertia of the cylinder about a vertical
axis is
2
0.025 kg m
and neglecting the effect of friction, determine
the angular velocity of the assembly as end B of the rod strikes end E
of the cylinder.
SOLUTION
Kinematics and geometry.
1
1.6 m/s
v
22
Initial position Final position
Conservation of angular momentum about C.
PROBLEM 17.83
A 1.6-kg tube AB can slide freely on rod DE, which in turn can rotate freely in a
horizontal plane. Initially the assembly is rotating with an angular velocity
5 rad/s
and the tube is held in position by a cord. The moment of inertia of
the rod and bracket about the vertical axis of rotation is
2
0.30 kg m
and the
centroidal moment of inertia of the tube about a vertical axis is
2
0.0025 kg m .
If
the cord suddenly breaks, determine (a) the angular velocity of the assembly after
the tube has moved to end E, (b) the energy lost during the plastic impact at E.
SOLUTION
Let Point C be the intersection of axle C and rod DE. Let Point G be the mass center of tube AB.
Masses and moments of inertia about vertical axes.
/
GGC
Syst. Momenta
1
Syst. Ext. Imp.
1
2
Syst. Momenta
2
PROBLEM 17.83 (Continued)
Moments about C:
11 1/1 2 2 2/2
22
/1 1 /2 2
22
2
2
()( ) 0 ()( )
() ()
[0.0025 0.30 (1.6)(0.0625) ](5) [0.0025 0.30 (1.6)(0.4375) ]
(0.30875)(5) 0.60875
AB DCE AB G C AB DCE AB G C
AB DCE AB G G AB DCE AB G C
II mvr I I mvr
I I mr I I mr
22 2
AB DCE AB
2222
3.859375 J
111
PROBLEM 17.84
In the helicopter shown, a vertical tail propeller is used to prevent
rotation of the cab as the speed of the main blades is changed.
Assuming that the tail propeller is not operating, determine the final
angular velocity of the cab after the speed of the main blades has
been changed from 180 to 240 rpm. (The speed of the main blades is
measured relative to the cab, and the cab has a centroidal moment of
inertia of 2
650 lb ft s .
Each of the four main blades is assumed to be
a slender 14-ft rod weighting 55 lb.)
2
3 32.2
21
2
(446.38)(8 6 )
446.38 650
2.5581 rad/s
B
CB
II
224.4 rpm
PROBLEM 17.85
Assuming that the tail propeller in Problem 17.84 is operating and
that the angular velocity of the cab remains zero, determine the final
horizontal velocity of the cab when the speed of the main blades is
changed from 180 to 240 rpm. The cab weighs 1250 lb and is initially
at rest. Also determine the force exerted by the tail propeller if the
change in speed takes place uniformly in 12 s.
2
446.38 lb ft s
The cab does not rotate. 12
0
112 2
S
y
st. Momenta S
y
st. Ext. Im
p
.S
y
st. Momenta
12
PROBLEM 17.86
The circular platform A is fitted with a rim of 200-mm inner radius and can
rotate freely about the vertical shaft. It is known that the platform-rim unit has
a mass of 5 kg and a radius of gyration of 175 mm with respect to the shaft.
At a time when the platform is rotating with an angular velocity of 50 rpm, a
3-kg disk B of radius 80 mm is placed on the platform with no velocity.
Knowing that disk B then slides until it comes to rest relative to the platform
against the rim, determine the final angular velocity of the platform.
0.7436(50 rpm)
2
PROBLEM 17.87
The 30-kg uniform disk A and the bar BC are at rest and the 5-kg
uniform disk D has an initial angular velocity 1
ω of magnitude 440
rpm when the compressed spring is released and disk D contacts
disk A. The system rotates freely about the vertical spindle BE.
After a period of slippage, disk D rolls without slipping. Knowing
that the magnitude of the final angular velocity of disk D is 176
rpm, determine the final angular velocities of bar BC and disk A.
Neglect the mass of bar BC.
SOLUTION
Data: 0.3 m, 0.115 m
AD
rr
2
22
11
30 0.3 1.35 kg m
AAA
Imr
2
22
11
PROBLEM 17.87 (Continued)
Kinetics.
PROBLEM 17.88
The 4-kg rod AB can slide freely inside the 6-kg tube CD. The rod
was entirely within the tube (x 0) and released with no initial
velocity relative to the tube when the angular velocity of the
assembly was 5 rad/s. Neglecting the effect of friction, determine
the speed of the rod relative to the tube when x 400 mm.
PROBLEM 17.88 (Continued)
State 1. 0,x 15rad/s,
0
r
v
11
Conservation of angular momentum: 12
() ()
OO
HH
2
10.6667 4.05333
22.6316 rad/s
r
r
PROBLEM 17.89
A 1.8-kg collar A and a 0.7-kg collar B can slide without friction
on a frame, consisting of the horizontal rod OE and the vertical
rod CD, which is free to rotate about its vertical axis of
symmetry. The two collars are connected by a cord running
over a pulley that is attached to the frame at O. At the instant
shown, the velocity
A
v of collar A has a magnitude of 2.1 m/s
and a stop prevents collar B from moving. The stop is suddenly
removed and collar A moves toward E. As it reaches a distance
of 0.12 m from O, the magnitude of its velocity is observed to
be 2.5 m/s. Determine at that instant the magnitude of the
angular velocity of the frame and the moment of inertia of the
frame and pulley system about CD.
2
PROBLEM 17.89 (Continued)
OAAA
22 2 2 2
( ) (1.8)(0.12 )(0.12) ( ) ( 0.02592)
OO
HI H I
(8)
11 1
