170 Solutions Manual
Problem 2.116
With reference to Probs. 2.113 and 2.115, assume that an experiment
is conducted so that the measured value of
xI
is 10% smaller than
what is predicted in the absence of viscous drag. Find the value of
⌘
that would be required for the theory in Prob. 2.113 to match the
experiment.
Solution
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Dynamics 2e 171
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172 Solutions Manual
Problem 2.117
Consider the vectors EaD2O{C1O|C7O
kand E
bD1O{C2O|C3O
k. Compute the following quantities.
(a) Ea⇥E
b
(b) E
b⇥Ea
(c) Ea⇥E
bCE
b⇥Ea
(d) Ea⇥Ea
(e) .Ea⇥Ea/ ⇥E
b
(f) Ea⇥.Ea⇥E
b/
Parts (a)–(d) of this problem are meant to be a reminder that the cross product is an anticommutative
operation, while Parts (e) and (f) are meant to be a reminder that the cross product is an operation that is
not associative.
Solution
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Dynamics 2e 173
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174 Solutions Manual
Problem 2.118
Consider two vectors EaD1O{C2O|C3O
kand E
bD6O{C3O|.
(a) Verify that Eaand E
bare perpendicular to one another.
(b) Compute the vector triple product Ea⇥.Ea⇥E
b/.
(c) Compare the result from calculating Ea⇥.Ea⇥E
b/ with the vector jEaj2E
b.
The purpose of this exercise is to show that as long as
Ea
and
E
b
are perpendicular to one another, you can
always write
Ea⇥.Ea⇥E
b/ DjEaj2E
b
. This identity turns out to be very useful in the study of the planar
motion of rigid bodies.
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Dynamics 2e 175
Problem 2.119
Let
Er
be the position vector of a point
P
with respect to a Cartesian coordinate system with axes
x
,
y
,
and
´
. Let the motion of
P
be confined to the
xy
plane, so that
ErDrxO{CryO|
(i.e.,
ErO
kD0
). Also,
let
E!rD!rO
k
be the angular velocity vector of the vector
Er
. Compute the outcome of the products
E!r⇥.E!r⇥Er/ and E!r⇥.Er⇥E!r/.
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176 Solutions Manual
Problem 2.120
The three propellers shown are all rotating with the same angular
speed of 1000 rpm about different coordinate axes.
(a)
Provide the proper vector expressions for the angular velocity
of each of the three propellers.
(b)
Suppose that an identical propeller rotates at
1000 rpm
about
the axis
`
oriented by the unit vector
Ou`
. Let any point
P
on
`
have coordinates such that
xPDyPD´P
. Find the vector
representation of the angular velocity of this fourth propeller.
Express the answers using units of radians per second.
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permission of McGraw-Hill, is prohibited.
Dynamics 2e 177
Problem 2.121
Point
P
is constrained to move along a straight line
`
whose positive orientation is described by the unit
vector
Ou`
. Point
A
is a fixed reference point on
`
. Let the vector
ErP=A
denote the position of
P
relative to
A
and let
OuP=A
be a unit vector pointing from
A
to
P
. Use the concept of time derivative of a vector to
describe the velocity and acceleration of
P
. In addition, comment on what happens to the description of
the velocity and acceleration when Phappens to coincide with the fixed point A.
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permission of McGraw-Hill, is prohibited.
Problem 2.122
Starting with Eq. (2.48), show that the second derivative with respect to time of an arbitrary vector
E
A
is
given by R
E
ADR
AOuAC2E!A⇥P
AOuACP
E!A⇥E
ACE!A⇥E!A⇥E
A:
Keep the answer in pure vector form, and do not resort to using components in any component system.
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permission of McGraw-Hill, is prohibited.
Dynamics 2e 179
Problem 2.123
The propeller shown has a diameter of
38 ft
and is rotating with
a constant angular speed of
400 rpm
. At a given instant, a point
P
on the propeller is at
ErPD.12:5 O{C14:3 O|/ft
. Use Eq. (2.48)
and the equation derived in Prob. 2.122 to compute the velocity
and acceleration of P; respectively.
Solution
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permission of McGraw-Hill, is prohibited.