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260 Solutions Manual
Problem 2.196
A micro spiral pump consists of a spiral channel attached to a stationary plate. This plate has two ports,
one for fluid inlet and another 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 the stationary
plate is put in motion by the rotation of the top disk, which pulls the fluid through the spiral channel.
Consider a spiral channel with the geometry given by the equation
rD⌘✓ Cr0
, where
r0D146 m
is the starting radius,
r
is the distance from the spin axis, and
✓
, measured in radians, is the angular position
of a point in the spiral channel. Assume that the radius at the outlet is
rout D190 m
, that the top disk
rotates with a constant angular speed
!
, and that the fluid particles in contact with the rotating disk are
essentially stuck to it. Determine the constant
⌘
and the value of
!
(in rpm) such that after
1:25 rev
of the
top disk, the speed of the particles in contact with this disk is vD0:5 m=s at the outlet.
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.
Dynamics 2e 261
Problem 2.197
A micro spiral pump consists of a spiral channel attached to a stationary plate. This plate has two ports,
one for fluid inlet and another 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 the stationary
plate is put in motion by the rotation of the top disk, which pulls the fluid through the spiral channel.
Consider a spiral channel with the geometry given by the equation
rD⌘✓ Cr0
, where
⌘D12 m
is called the polar slope,
r0D146 m
is the starting radius,
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, use the polar coordinate system shown and determine the velocity and
acceleration of one fluid particle when it is at rD170 m.
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.
262 Solutions Manual
Problem 2.198
The cutaway of the gun barrel shows a projectile that, upon exit, moves with a speed
vsD5490 ft=s
relative to the gun barrel. The length of the gun barrel is
LD15 ft
. Assuming that the angle
✓
is increasing
at a constant rate of
0:15 rad=s
, determine the speed of the projectile right when it leaves the barrel. In
addition, assuming that the projectile acceleration along the barrel is constant and that the projectile starts
from rest, determine the magnitude of the acceleration upon exit.
Solution
Using the polar coordinate system shown at the right, the velocity
Dynamics 2e 263
Problem 2.199
A particle moves along a spiral described by the equation
rDr0C✓
,
where
r0
and
are constants, and where
✓
is in radians. Assume that
P
✓D˛t
, where
˛D0:15 rad=s2
and
t
is time expressed in seconds.
If
rD0:25
m and
✓D0
for
tD0
, determine
such that, for
tD10
s, the acceleration is completely in the radial direction. In
addition, determine the value of the polar coordinates of the point for
tD10 s.
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.
264 Solutions Manual
Problem 2.200
A point is moving counterclockwise at constant speed
v0
along a spiral
described by the equation
rDr0C✓
, where
r0
and
are constants
with dimensions of length. Determine the expressions of the velocity
and the acceleration of the particle as a function of
✓
expressed in the
polar component system shown.
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 265
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.
266 Solutions Manual
Problem 2.201
A person driving along a rectilinear stretch of road is fined for speeding, having been clocked at
75 mph
when the radar gun was pointing as shown. The driver claims that, because the radar gun is off to the side
of the road instead of directly in front of his car, the radar gun overestimates his speed. Is he right or wrong
and why?
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 267
Problem 2.202
A motion tracking camera is placed along a rectilinear stretch of a
racetrack (the figure is not to scale). A car
C
enters the stretch at
A
with a speed
vAD110 mph
and accelerates uniformly in time so
that at
B
it has a speed
vBD175 mph
, where
dD1mi
. Letting
the distance
LD50 ft
, if the camera is to track
C
, determine the
camera’s angular velocity and the time rate of change of the angular
velocity when the car is at Aand at H.
Solution
Referring to figure at the right, the camera points from
O
to the car
along the direction of
Our
. Hence, the angular velocity and acceleration
268 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 269
Problem 2.203
The radar station at
O
is tracking a meteor
P
as it moves through the
atmosphere. At the instant shown, the station measures the following data
for the motion of the meteor:
rD21;000 ft
,
✓D40ı
,
PrD22;440 ft=s
,
P
✓D2:935 rad=s, RrD187;500 ft=s2, and R
✓D5:409 rad=s2.
(a)
Determine the magnitude and direction (relative to the
xy
coordinate
system shown) of the velocity vector at this instant.
(b)
Determine the magnitude and direction (relative to the
xy
coordinate
system shown) of the acceleration vector at this instant.
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.
