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190 Solutions Manual
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 191
Problem 2.134
A micro spiral pump consists of a spiral channel attached to a stationary plate. This plate has two ports,
one for fluid inlet and the other for outlet, the outlet being farther from the center of the plate than the
inlet. The system is capped by a rotating disk. The fluid trapped between the rotating disk and stationary
plate is put in motion by the rotation of the top disk, which pulls the fluid through the spiral channel. With
this in mind, consider a channel with geometry given by the equation
rD⌘✓ Cr0
, where
⌘D12
m is
called the polar slope,
r0D146
m is the radius at the inlet,
r
is the distance from the spin axis, and
✓
,
measured in radians, is the angular position of a point in the spiral channel. If the top disk rotates with a
constant angular speed
!D30;000 rpm
, and assuming that the fluid particles in contact with the rotating
disk are essentially stuck to it, determine the velocity and acceleration of one such fluid particle when it is
at
rD170
m. Express the answer using the component system shown (which rotates with the top disk).
Photo credit: “Design and Analysis of a Surface Micromachined Spiral-Channel Viscous Pump,” by M. I. Kilani, P. C. Galambos, Y. S. Haik,
C. H. Chen, Journal of Fluids Engineering, Vol. 125, pp. 339–344, 2003.
Solution
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.
192 Solutions Manual
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 193
Problem 2.135
A disk rotates about its center, which is the fixed point
O
. The disk has
a straight channel whose centerline passes by
O
and within which a collar
A
is allowed to slide. If, when
A
passes by
O
, the speed of
A
relative to
the channel is
vD14 m=s
and is increasing in the direction shown with a
rate of
5m=s2
, determine the acceleration of
A
given that
!D4rad=s
and
is constant. Express the answer using the component system shown, which
rotates with the disk. Hint: Apply the equation derived in Prob. 2.122 to the
vector describing the position of Arelative to Oand then let rD0.
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.
194 Solutions Manual
Problem 2.136
At the instant shown, the angular velocity and acceleration of the
merry-go-round are as indicated in the figure. The distance of
the child from the spin axis is
rP
, so his acceleration is
EaPD
RrPOurCPrPP
OurCP
E!⇥rPOurCE!⇥PrPOurCE!⇥rPP
Our
. Assuming
that the child is walking along a radial line, should the child walk
outward or inward to make sure that he does not experience any
sideways acceleration (i.e., in the direction of Ouq)?
Solution
Using the component system shown, the position of the child is
ErPDrOur
, where
r
is the distance from
the spin axis and
Our
is the unit vector always pointing from the origin of the system toward
P
. Applying
Dynamics 2e 195
Problem 2.137
Assuming that the child shown is moving on the merry-go-round
along a radial line, use the equation derived in Prob. 2.122 to
determine the relation that
!
,
P!
,
r
, and
Pr
must satisfy so that the
child will not experience any sideways acceleration.
Solution
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.
196 Solutions Manual
Problem 2.138
The mechanism shown is called a swinging block slider crank. First used in various steam locomotive
engines in the 1800s, this mechanism is often found in door-closing systems. If the disk is rotating with
a constant angular velocity
P
✓D60 rpm
,
HD4ft
,
RD1:5 ft
, and
r
is the distance between
B
and
O
,
compute
Pr
and
P
when
✓D90ı
.Hint: Apply Eq. (2.48) to the vector describing the position of
B
relative
to O.
Solution
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.
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.
198 Solutions Manual
Problem 2.139
A sprinkler essentially consists of a pipe
AB
mounted on a hollow shaft. The water comes in the pipe
at
O
and goes out the nozzles at
A
and
B
, causing the pipe to rotate. Assume that the particles of water
move through the pipe at a constant rate relative to the pipe of
5ft=s
and that the pipe
AB
is rotating
at a constant angular velocity of
250 rpm
. In all cases, express the answers using the right-handed and
orthogonal component system shown.
Determine the acceleration of the water particles when they are at
d=2
from
O
(still within the
horizontal portion of the pipe). Let dD7in:
Solution
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 199
Problem 2.140
A sprinkler essentially consists of a pipe
AB
mounted on a hollow shaft. The water comes in the pipe
at
O
and goes out the nozzles at
A
and
B
, causing the pipe to rotate. Assume that the particles of water
move through the pipe at a constant rate relative to the pipe of
5ft=s
and that the pipe
AB
is rotating
at a constant angular velocity of
250 rpm
. In all cases, express the answers using the right-handed and
orthogonal component system shown.
Determine the acceleration of the water particles right before they are expelled at
B
. Let
dD7in:
,
ˇD15ı
, and
LD2in:
Hint: In this case, the vector describing the position of a water particle at
B
goes
from Oto Band is best written as ErDrBOuBCr´O
k.
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.
