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60 Solutions Manual
Problem 2.27
Find the
x
and
y
components of the acceleration in Example 2.3 (except for the plots) by differentiating
the first of Eqs. (3) and the last of Eqs. (1) with respect to time and then solving the resulting two equations
for Rxand Ry. Verify that you get the results given in Example 2.3.
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 61
Problem 2.28
Airplane
A
is performing a loop with constant radius
⇢D1000 ft
. The equation describing the loop is as
follows:
.x xC/2C.y yC/2D⇢2;
where
xCD0
and
yCD1500 ft
are the coordinates of the center of the loop. If the plane were capable of
maintaining its speed constant and equal to
v0D160 mph
, determine the velocity and acceleration of the
plane for ✓D30ı.
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.
62 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 63
Problem 2.29
An airplane
A
takes off as shown with a constant speed equal to
v0D160 km=h
. The path of the airplane is described by the equa-
tion
yDx2
, where
D6⇥104m1
. Using the component
system shown, provide the expression for the velocity and accelera-
tion of the airplane when
xD400
m. Express the velocity in
m=s
and the acceleration in m=s2.
Solution
The position of the airplane can be described as
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.
64 Solutions Manual
Problem 2.30
A test track for automobiles has a portion with a specific profile described by:
yDh⇥1sin.x=w/⇤;
where
hD0:5 ft
and
wD8ft
, and where the argument of the sine function is understood to be in radians.
A car travels in the positive
x
direction such that the horizontal component of velocity remains constant
and equal to
55 mph
. Modeling the car as a point moving along the given profile, determine the maximum
speed of the car. Express your answer in ft=s.
Solution
Letting xand yrepresent the coordinates of the car, the position of the car is
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 65
Problem 2.31
A test track for automobiles has a portion with a specific profile described by:
yDh⇥1cos.x=w/⇤;
where
hD0:20
m and
wD2
m, and where the argument of the cosine function is understood to be
in radians. A car travels in the positive
x
direction with a constant
x
component of velocity equal
to
100 km=h
. Modeling the car as a point moving along the given profile, determine the velocity and
acceleration (expressed in m=s and m=s2, respectively) of the car for xD24 m.
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.
66 Solutions Manual
Problem 2.32
A test track for automobiles has a portion with a specific profile described by:
yDh⇥1cos.x=w/⇤;
where
hD0:75 ft
and
wD10 ft
, and where the argument of the cosine function is understood to be in
radians. A car drives at a constant speed
v0D35 mph
. Modeling the car as a point moving along the given
profile, find the velocity and acceleration of the car for
xD97 ft
. Express velocity in
ft=s
and acceleration
in ft=s2.
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 67
Differentiating the last of Eqs. (3) with respect to xwe have
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.
68 Solutions Manual
Problem 2.33
The orbit of a satellite
A
around planet
B
is the ellipse shown and is described by the equation
.x=a/2C
.y=b/2D1
, where
a
and
b
are the semimajor and semiminor axes of the ellipse, respectively. When
xDa=2
and
y>0
, the satellite is moving with a speed
v0
as shown. Determine the expression for the
satellite’s velocity Evin terms of v0,a, and bfor xDa=2 and y>0.
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 69
Problem 2.34
In the mechanism shown, block Bis fixed and has a profile described by the following relation:
yDh"1C1
2✓x
d◆2
1
4✓x
d◆4#:
The follower moves with the shuttle A, and the tip Cof the follower remains in contact with B.
Assume that
hD0:25 in:
,
dD1in:
, and the horizontal position of
C
is
xDdsin.!t/
, where
!D2⇡rad=s
, and
t
is time in seconds. Determine an analytical expression for the speed of
C
as a
function of
x
and the parameters
d
,
h
, and
!
. Then, evaluate the speed of
C
for
xD0
,
xD0:5 in:
, and
xD1in. Express your answers in ft=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.
