Dynamics 2e 2189
Problem 10.58
The uniform sphere of mass
m
and radius
R
can spin freely relative to
the shaft
AB
, whose mass is negligible. The shaft
AB
precesses about
the vertical axis with constant angular speed
!0
. Assuming that the
friction between the sphere and the horizontal surface above is sufficient
to prevent slipping and neglecting friction in the pin at
A
, determine an
expression for the minimum value of
!0
for which this motion is possible.
Solution
The FBD of the sphere and shaft are shown at the right, where
Ft
is the friction force between the sphere and the surface
on which the sphere is rolling. The
xy´
frame has its origin
at
A
, it rotates with
!0
, and its
x
axis is aligned with the
shaft
AB
. The
x0y0´0
frame has its origin at
A
and it also
rotates with
!0
. The planes defined
xy
and
x0y0
are always
coincident.
!1
is the angular speed of the sphere relative to
the shaft.
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2190 Solutions Manual
where
C
is the point on the sphere in contact with the surface above it and
E!s
is the angular velocity of the
sphere. Writing O|0in terms of the xy´ frame and expanding cross products, we obtain
2R!0O
kDŒ!0.sin ✓O{Ccos ✓O|/C!1O{ç⇥ŒR.sin ✓O{Ccos ✓O|/çDR!1cos ✓O
k
)!1D2!0
cos ✓:(6)
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Dynamics 2e 2191
Problem 10.59
The bar
AB
rotates about the vertical
y
axis with angular speed
!b.t/
. The disk, whose center is at
C
, can rotate freely relative to
the arm CD as it rolls without slipping on the horizontal surface.
If
!b
is constant, determine the angular velocity
E!d
of the
disk. Express your answer in the rotating
xy´
component system
shown.
Solution
Using the given xy´ frame, the angular velocity of the disk is
E!dD!bO|CP
ˇO
k: (1)
To find
P
ˇ
, we relate the velocity of
C
to the velocity of the point of contact between the disk and the surface,
which we will call Q,
EvCDEvQCE!d⇥ErC=Q )h!bO{D⇣!bO|CP
ˇO
k⌘⇥RO|DRP
ˇO{)P
ˇDh
R!b:(2)
2192 Solutions Manual
Problem 10.60
The bar
AB
rotates about the vertical
y
axis with angular speed
!b.t/
. The disk, whose center is at
C
, can rotate freely relative to
the arm CD as it rolls without slipping on the horizontal surface.
If
!b.t/
is a known function of time, determine the angular
acceleration
E˛d
of the disk. Express your answer in the rotating
xy´ component system shown.
Solution
Using the given xy´ frame, the angular velocity of the disk is
E!dD!bO|CP
ˇO
k: (1)
Since !bis not constant, the angular acceleration of the disk is
E˛dDP!bO|CR
ˇO
kCP
ˇ⇣!bO|⇥O
k⌘DP
ˇ!bO{CP!bO|CR
ˇO
k: (2)
To find
P
ˇ
and
R
ˇ
, we relate the velocity of
C
to the velocity of the point of contact between the disk and the
surface, which we will call Q,
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Dynamics 2e 2193
Problem 10.61
The bar
AB
rotates about the vertical
y
axis with angular speed
!b.t/
. The disk, whose center is at
C
, can rotate freely relative to
the arm CD as it rolls without slipping on the horizontal surface.
If
!b
is constant, determine the velocity
EvP
of the point
P
on
the periphery of the disk as a function of the angle
ˇ
. Express
your answer in the rotating xy´ component system shown.
Solution
To find the velocity of
P
, we can relate it to the velocity of
C
, which can be easily found. In doing so, we
will also need the angular velocity of the disk.
Relating the velocity of Pto that of C, we obtain
EvPDEvCCE!d⇥ErP=C;(1)
where, using the given xy´ frame, the angular velocity of the disk is
E!dD!bO|CP
ˇO
k; (2)
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2194 Solutions Manual
Problem 10.62
The bar
AB
rotates about the vertical
y
axis with angular speed
!b.t/
. The disk, whose center is at
C
, can rotate freely relative to
the arm CD as it rolls without slipping on the horizontal surface.
If
!b.t/
is a known function of time, determine the accelera-
tion
EaP
of the point
P
on the periphery of the disk as a function of
the angle
ˇ
. Express your answer in the rotating
xy´
component
system shown.
Solution
The acceleration of
P
can be found by relating it to the acceleration of point
C
, which is easily found. In
doing so, we will need to find the angular velocity and angular acceleration of the disk.
Relating the acceleration of Pto that of C, we obtain
EaPDEaCCE˛d⇥ErP=C CE!d⇥E!d⇥ErP=C;(1)
where, using the given xy´ frame, the acceleration of Cis
EaCDh!2
bO
k; (2)
the angular velocity of the disk is
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Dynamics 2e 2195
Problem 10.63
The shaft
AB
rotates with angular speed
!0.t/
about the vertical axis. The
uniform thin rod
CD
of length
L
is rigidly attached to the end of the hori-
zontal arm OG, which is rigidly attached to AB. The rod CD is tilted from
the vertical position through the angle
✓
in the
XY
plane about the
Z
axis.
Express your answers in the XYZ frame.
If the angular speed
!0
of the vertical shaft is constant, determine the
reaction at Oon the horizontal shaft required for this motion.
Solution
The FBD of the segment
OG
and the bar
CD
is
shown on the right, where we have neglected the
weight of the bar
OG
. Both the
xy´
and
XYZ
frames have their origin at
G
and they rotate with
!0.
Balance Principles.
Summing forces in the
X
,
Y
,
and Zdirections, we obtain
XFXWOXDmaGX ;(1)
XFYWOYmg DmaGY ;(2)
2196 Solutions Manual
The position of Grelative to Ois
ErG=O DLO
IDL.cos ✓O{Csin ✓O|/:(9)
The angular velocity and angular acceleration of the bar CD are, respectively,
E!bD!0O
JD!0.sin ✓O{Ccos ✓O|/and P
E!bDE
0: (10)
Computation. Substituting Eqs. (??)–(??) into Eqs. (??)–(??), we obtain the following six equations
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Dynamics 2e 2197
Problem 10.64
The shaft
AB
rotates with angular speed
!0.t/
about the vertical axis. The
uniform thin rod
CD
of length
L
is rigidly attached to the end of the hori-
zontal arm OG, which is rigidly attached to AB. The rod CD is tilted from
the vertical position through the angle
✓
in the
XY
plane about the
Z
axis.
Express your answers in the XYZ frame.
If the angular speed
!0.t/
of the vertical shaft is not constant, determine
the reaction at Oon the horizontal shaft required for this motion.
Solution
The FBD of the segment
OG
and the bar
CD
is
shown on the right, where we have neglected the
weight of the bar
OG
. Both the
xy´
and
XYZ
frames have their origin at
G
and they rotate with
!0.
Balance Principles.
Summing forces in the
X
,
Y
,
and Zdirections, we obtain
XFXWOXDmaGX ;(1)
XFYWOYmg DmaGY ;(2)
XFZWOZDmaGZ:(3)
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2198 Solutions Manual
where we have used the fact that O
KDO
k. The position of Grelative to Ois
ErG=O DLO
IDL.cos ✓O{Csin ✓O|/:(9)
The angular velocity and angular acceleration of the bar CD are, respectively,
E!bD!0O
JD!0.sin ✓O{Ccos ✓O|/and P
E!bDP!0O
JDP!0.sin ✓O{Ccos ✓O|/:(10)
Computation. Substituting Eqs. (??)–(??) into Eqs. (??)–(??), we obtain the following six equations
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