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360 Solutions Manual
Problem 2.278
An airplane is being tracked by a radar station at
A
. At the instant
tD0
, the following data is recorded:
rD15 km
,
D80ı
,
✓D15ı
,
PrD350 km=h
,
P
D0:002 rad=s
,
P
✓D0:003 rad=s
.
If the airplane is flying to keep each of the spherical velocity
components constant for a few minutes, determine the spherical
components of the airplane’s acceleration when tD30 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.
Dynamics 2e 361
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.
362 Solutions Manual
Problem 2.279
An airplane is being tracked by a radar station at
A
. At the instant
tD0
, the following data is recorded:
rD15 km
,
D80ı
,
✓D
15ı
,
PrD350 km=h
,
P
D0:002 rad=s
,
P
✓D0:003 rad=s
. If the
airplane is flying to keep each of the spherical velocity components
constant, plot the trajectory of the airplane for 0 < t < 150 s.
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 363
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.
364 Solutions Manual
Problem 2.280
The velocity and acceleration of point Pexpressed relative to frame Aat some time tare
EvP=A D.12:5 O{AC7:34 O|A/m=s and EaP=A D.7:23 O{A3:24 O|A/m=s2:
Knowing that frame
B
does not move relative to frame
A
, determine the expressions for the velocity
and acceleration of
P
with respect to frame
B
. Verify that the speed of
P
and the magnitude of
P
’s
acceleration are the same in the two frames.
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 365
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.
366 Solutions Manual
Problem 2.281
The motion of a point
P
with respect to a Cartesian coordinate system is described
by
ErDf2ptO{CŒ4 ln.t C1/ C2t2çO|gft
, where
t
is time expressed in seconds.
Determine the average velocity between
t1D4
s and
t2D6
s. Then find the time
N
t
for which the
x
component of
P
’s velocity is exactly equal to the
x
component
of
P
’s average velocity between times
t1
and
t2
. Is it possible to find a time at
which
P
’s velocity and
P
’s average velocity are exactly equal? Explain why. Hint:
Velocity is a vector.
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 367
Problem 2.282
The figure shows the displacement vector of a point
P
between two
time instants
t1
and
t2
. Is it possible for the vector
Evavg
shown to be the
average velocity of Pover the time interval Œt1;t
2ç?
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.
368 Solutions Manual
Problem 2.283
A dynamic fracture model proposed to explain the behavior of cracks
propagating at high velocity views the crack path as a wavy path.
In this model, a crack tip appearing to travel along a straight path
actually travels at roughly the speed of sound along a wavy path.
Let the wavy path of the crack tip be described by the function
yDhsin.2⇡x=/
, where
h
is the amplitude of the crack tip fluctu-
ations in the direction perpendicular to the crack plane and
is the
corresponding period. Assume that the crack tip travels along the
wavy path at a constant speed vs(e.g., the speed of sound).
Find the expression for the
x
component of the crack tip velocity
as a function of vs,,h, and x.
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 369
Problem 2.284
A dynamic fracture model proposed to explain the behavior of cracks
propagating at high velocity views the crack path as a wavy path.
In this model, a crack tip appearing to travel along a straight path
actually travels at roughly the speed of sound along a wavy path.
Let the wavy path of the crack tip be described by the function
yDhsin.2⇡x=/
, where
h
is the amplitude of the crack tip fluctu-
ations in the direction perpendicular to the crack plane and
is the
corresponding period. Assume that the crack tip travels along the
wavy path at a constant speed vs(e.g., the speed of sound).
Denote the apparent crack tip velocity by
va
, and define it as the
average value of the xcomponent of the crack velocity, that is,
vaD1
Z
0
vxdx:
In dynamic fracture experiments on polymeric materials,
vaD
2vs=3
,
vs
is found to be close to
800 m=s
, and
is of the order
of 100
m
. What value of
h
would you expect to find in the experi-
ments if the wavy crack theory were confirmed to be accurate?
Solution
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permission of McGraw-Hill, is prohibited.
