Dynamics 2e 21
Problem 1.16
Two Coast Guard patrol boats
P1
and
P2
are stationary while monitoring the motion of a surface vessel
A
.
The velocity of Awith respect to P1is expressed by
EvAD.23 O{16O|1/ft=s;
whereas the acceleration of A, expressed relative to P2, is given by
EaAD.2O{24O|2/ft=s2:
Determine the velocity and the acceleration of
A
expressed with respect to the land-based component
system .O{; O|/.
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permission of McGraw-Hill, is prohibited.
22 Solutions Manual
Problem 1.17
The measure of angles in radians is defined according to the following relation:
r DsAB
, where
r
is the
radius of the circle and
sAB
denotes the length of the circular arc. Determine the dimensions of the angle
.
Solution
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permission of McGraw-Hill, is prohibited.
Dynamics 2e 23
Problem 1.18
Letting
C
denote the circumference of a circle, a
1ı
angle is, by definition, an angle that subtends an arc of
length
`
such that
C=` D360
. Apply the definition of degree and determine the radius of the circle shown
knowing that the length sof the arc subtended by the 4ıangle in the figure is 1:84 mm.
Solution
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permission of McGraw-Hill, is prohibited.
24 Solutions Manual
Problem 1.19
A simple oscillator consists of a linear spring fixed at one end and a mass attached at the other
end, which is free to move. Suppose that the periodic motion of a simple oscillator is described
by the relation
yDY0sin.2!0t/
, where
y
has units of length and denotes the vertical
position of the oscillator,
Y0
is the oscillation amplitude,
!0
is the oscillation frequency, and
t
is time. Recalling that the argument of a trigonometric function is an angle, determine the
dimensions of
Y0
and
!0
, as well as their units in both the SI and the U.S. Customary systems.
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 25
Problem 1.20
To study the motion of a space station, the station can be modeled as a rigid
body and the equations describing its motion can be chosen to be Euler’s
equations, which read
MxDIxx ˛xIyy I´´!y!´;
MyDIyy ˛yI´´ Ixx!x!´;
M´DI´´ ˛´Ixx Iyy !x!y:
In the previous equations,
Mx
,
My
, and
M´
denote the
x
,
y
, and
´
com-
ponents of the moment applied to the body;
!x
,
!y
, and
!´
denote the
corresponding components of the angular velocity of the body, where angu-
lar velocity is defined as the time rate of change of an angle;
˛x
,
˛y
, and
˛´
denote the corresponding components of the angular acceleration of the
body, where angular acceleration is defined as the time rate of change of an
angular velocity. The quantities
Ixx
,
Iyy
, and
I´´
are called the principal
mass moments of inertia of the body. Determine the dimensions of
Ixx
,
Iyy
,
and
I´´
and determine their units in SI, as well as in the U.S. Customary
system.
y
z
ជ
M
ជ
␣ជ
y
␣
Solution
Let
L
,
M
, and
T
denote length, mass, and time, respectively. We consider only one of the three equations as
their dimensions are identical:
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permission of McGraw-Hill, is prohibited.
26 Solutions Manual
Problem 1.21
The lift force FLgenerated by the airflow moving over a wing is often expressed as follows:
FLD1
2v2CL./A; (1)
where
,
v
, and
A
denote the mass density of air, the airspeed (relative to the wing), and the wing’s nominal
surface area, respectively. The quantity
CL
is called the lift coefficient, and it is a function of the wing’s
angle of attack . Find the dimensions of CLand determine its units in the SI system.
Solution
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permission of McGraw-Hill, is prohibited.
Dynamics 2e 27
Problem 1.22
Are the words units and dimensions synonyms?
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permission of McGraw-Hill, is prohibited.
28 Solutions Manual
Problem 1.23
A rock is released from rest into water. The magnitude
Fd
of the drag force acting on the rock due to its
motion through water can be modeled as
FdDCdv
, where
v
is the speed of the rock and
Cd
is a constant
drag coefficient. Determine the units used to measure Cdin the U.S. Customary system.
Solution
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permission of McGraw-Hill, is prohibited.
Dynamics 2e 29
Problem 1.24
In elementary beam theory, for a uniform beam supported as shown, the relation between the force
P
applied at the end of a beam and the corresponding end deflection
ı
is
PD.3EI =L3/ı
, where
E
is a
constant called the modulus of elasticity,
I
is a constant called the centroidal area moment of inertia, and
L
is the length of the beam. If the dimensions of
I
are length to the power four, determine the SI units
used to measure the constant E.
Solution
Let `,m, and tdenote length, mass, and time, respectively. Since Pis a force, its dimensions are
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