978-0073380308 Chapter 2 Solution Manual Part 24

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

,

PrD22;440 ft=s

,

P

✓D2:935 rad=s, RrD187;500 ft=s2, and R

✓D5:409 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.