978-0073380308 Chapter 2 Solution Manual Part 34

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

D0:002 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

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Dynamics 2e 361

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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

D0:002 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.

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permission of McGraw-Hill, is prohibited.

Dynamics 2e 363

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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{A3:24 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

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permission of McGraw-Hill, is prohibited.

Dynamics 2e 365

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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

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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?

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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.

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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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