Dynamics 2e 2119
˛ABy Dhh2L2
AB R2!2
d
L2
AB L2
AB d2h23=2 ;
AB R2!2
d
of McGraw-Hill, and must be surrendered upon request of McGraw-Hill. Any duplication or distribution, either in print or electronic form, without the
permission of McGraw-Hill, is prohibited.
2120 Solutions Manual
Problem 10.29
Bar
AB
of length
LAB D2:5
m is attached by ball
joints to a collar at
A
and to a disk at
B
. The disk
lies in the
xy
plane and its center at
E
lies on the
y
axis in the
y´
plane. The disk rotates about a vertical
axis at the constant angular rate
!dD100 rpm
. The
dimensions
dD1:2
m,
hD0:9
m, and
RD0:75
m
are given.
For the disk position
✓D0ı
, determine the ve-
locity of the collar at
A
. Express your answer in
the given component system, and assume that the
angular velocity of the bar is orthogonal to it.
Solution
To determine the velocity of the collar
A
, we will relate the velocities of the ends of the bar to one another
and then use the constraint that the angular velocity is orthogonal to the bar to make it unique. Relating the
velocity of Ato that of B, we obtain
EvADEvBCE!AB ⇥ErA=B ;(1)
where, since the collar Ais constrained to move only in the xdirection, we have that
EvADvAO{:
In the position shown, the velocity of Bis
EvBDR!dO|;
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permission of McGraw-Hill, is prohibited.
Dynamics 2e 2121
Writing that the angular velocity of bar AB must be orthogonal to the bar, we obtain the fourth equation as
E!AB ErA=B D0).` R/!ABx d!
ABy Ch!AB´ D0:
Solving these four equations for the four unknowns listed above, we obtain
of McGraw-Hill, and must be surrendered upon request of McGraw-Hill. Any duplication or distribution, either in print or electronic form, without the
permission of McGraw-Hill, is prohibited.
2122 Solutions Manual
Problem 10.30
Bar
AB
of length
LAB D2:5
m is attached by ball
joints to a collar at
A
and to a disk at
B
. The disk
lies in the
xy
plane and its center at
E
lies on the
y
axis in the
y´
plane. The disk rotates about a vertical
axis at the constant angular rate
!dD100 rpm
. The
dimensions
dD1:2
m,
hD0:9
m, and
RD0:75
m
are given.
For the disk position
✓D0ı
, determine the ac-
celeration of the collar at
A
. Express your answer
in the given component system, and assume that the
angular velocity and angular acceleration of the bar
are orthogonal to it.
Solution
We need to first determine the angular velocity of the bar
AB
, which we can do by relating the velocities of
the ends of the bar to one another and then use the constraint that the angular velocity is orthogonal to the bar
to make it unique. Relating the velocity of Ato that of B, we obtain
EvADEvBCE!AB ⇥ErA=B ;(1)
where, since the collar Ais constrained to move only in the xdirection, we have that
EvADvAO{:
In the position shown, the velocity of Bis
EvBDR!dO|;
and the position of Arelative to Bis
of McGraw-Hill, and must be surrendered upon request of McGraw-Hill. Any duplication or distribution, either in print or electronic form, without the
permission of McGraw-Hill, is prohibited.
Dynamics 2e 2123
Writing that the angular velocity of bar AB must be orthogonal to the bar, we obtain the fourth equation as
E!AB ErA=B D0).` R/!ABx d!
ABy Ch!AB´ D0:
Solving these four equations for the four unknowns listed above, we obtain
!ABx DhR!d
d2Ch2C.`R/2;!
ABy DdhR!d
hd2Ch2C.`R/2i.` R/
;
!AB´ D
Rhd2C.`R/2i!d
hd2Ch2C.`R/2i.` R/
;v
ADdR!d
`R:
Finally, since the length of bar AB is LAB D2:5 m, we can find the length `using
of McGraw-Hill, and must be surrendered upon request of McGraw-Hill. Any duplication or distribution, either in print or electronic form, without the
permission of McGraw-Hill, is prohibited.
2124 Solutions Manual
˛ABy Dhh2L2
AB R2!2
d
L2
of McGraw-Hill, and must be surrendered upon request of McGraw-Hill. Any duplication or distribution, either in print or electronic form, without the
permission of McGraw-Hill, is prohibited.
Dynamics 2e 2125
Problem 10.31
Bar
AB
of length
LAB D2:5
m is attached by ball
joints to a collar at
A
and to a disk at
B
. The disk
lies in the
xy
plane and its center at
E
lies on the
y
axis in the
y´
plane. The disk rotates about a vertical
axis at the constant angular rate
!dD100 rpm
. The
dimensions
dD1:2
m,
hD0:9
m, and
RD0:75
m
are given.
Determine the angular velocity of the bar
AB
and
the velocity of the collar at
A
for any position
✓
of the
disk. Express your answers in the given component
system, and assume that the angular velocity of the
bar is orthogonal to it.
Solution
To determine the velocity of the collar
A
, we will relate the velocities of the ends of the bar to one another
and then use the constraint that the angular velocity is orthogonal to the bar to make it unique. Relating the
velocity of Ato that of B, we obtain
EvADEvBCE!AB ⇥ErA=B ;(1)
where, since the collar Ais constrained to move only in the xdirection, we have that
EvADvAO{:
For arbitrary ✓, the velocity of Bis
EvBD!dO
k⇥R.sin ✓O{cos ✓O|/DR!d.cos ✓O{Csin ✓O|/;
and the position of Arelative to Bis
of McGraw-Hill, and must be surrendered upon request of McGraw-Hill. Any duplication or distribution, either in print or electronic form, without the
permission of McGraw-Hill, is prohibited.
2126 Solutions Manual
Expanding cross products and equating components, we obtain the following three equations for the unknowns
!ABx,!ABy ,!AB´, and vA
vADh!ABy Cd!
AB´ R!AB´ cos ✓CR!dcos ✓;
0Dh!ABx C`!AB´ R!AB´ sin ✓CR!dsin ✓;
0Dd!
ABx `!ABy CR!ABx cos ✓CR!ABy sin ✓:
Writing that the angular velocity of bar AB must be orthogonal to the bar, we obtain the fourth equation as
E!AB ErA=B D0).`Rsin ✓/!ABx C.dCRcos ✓/!ABy Ch!AB´ D0:
Substituting in the expression found above for
`
and then solving these four equations for the four unknowns
listed above, we obtain
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of McGraw-Hill, and must be surrendered upon request of McGraw-Hill. Any duplication or distribution, either in print or electronic form, without the
permission of McGraw-Hill, is prohibited.
July 2, 2012
Dynamics 2e 2127
Problem 10.32
Bar
AB
of length
LAB D2:5
m is attached by ball
joints to a collar at
A
and to a disk at
B
. The disk
lies in the
xy
plane and its center at
E
lies on the
y
axis in the
y´
plane. The disk rotates about a vertical
axis at the constant angular rate
!dD100 rpm
. The
dimensions
dD1:2
m,
hD0:9
m, and
RD0:75
m
are given.
Determine the angular acceleration of the bar
AB
and the acceleration of the collar at
A
for any
position
✓
of the disk. Express your answers in the
given component system, and assume that the angu-
lar velocity and angular acceleration of the bar are
orthogonal to it.
Solution
We need to start by finding the angular velocity of bar
AB
for any
✓
, which we can do by relating the velocities
of the ends of the bar to one another and then use the constraint that the angular velocity is orthogonal to the
bar to make it unique. Relating the velocity of Ato that of B, we obtain
EvADEvBCE!AB ⇥ErA=B ;(1)
where, since the collar Ais constrained to move only in the xdirection, we have that
EvADvAO{:
For arbitrary ✓, the velocity of Bis
of McGraw-Hill, and must be surrendered upon request of McGraw-Hill. Any duplication or distribution, either in print or electronic form, without the
permission of McGraw-Hill, is prohibited.
2128 Solutions Manual
Expanding cross products and equating components, we obtain the following three equations for the unknowns
!ABx,!ABy ,!AB´, and vA
vADh!ABy Cd!
AB´ R!AB´ cos ✓CR!dcos ✓;
0Dh!ABx C`!AB´ R!AB´ sin ✓CR!dsin ✓;
0Dd!
ABx `!ABy CR!ABx cos ✓CR!ABy sin ✓:
Writing that the angular velocity of bar AB must be orthogonal to the bar, we obtain the fourth equation as
of McGraw-Hill, and must be surrendered upon request of McGraw-Hill. Any duplication or distribution, either in print or electronic form, without the
permission of McGraw-Hill, is prohibited.