Dynamics 2e 1923
Chapter 9 Solutions
Problem 9.1
Show that Eq. (9.15) is equivalent to Eq. (9.3) if CDpA2CB2and tan DA=B.
where we have use the identity that if
tan DA=B
, then
cos DB=pA2CB2
. Therefore, rewriting the
expression for x.t/ by replacing B= cos with C, we have
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Problem 9.2
Derive the formula for the mass moment of inertia of an arbitrarily shaped rigid body
about its mass center based on the body’s period of oscillation
when suspended as a
pendulum. Assume that the mass of the body
m
is known and that the location of the
mass center Gis known relative to the pivot point O.
42or IGDmgL2
42mL2;
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Dynamics 2e 1925
Problem 9.3
The thin ring of radius
R
and mass
m
is suspended by the pin at
O
. Determine its
period of vibration if it is displaced a small amount and released.
2R sin D0:
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Problem 9.4
The thin square hoop has mass
m
and is suspended by the pin at
O
. Determine its
period of vibration if it is displaced a small amount and released.
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Dynamics 2e 1927
Problem 9.5
The swinging bar and the vibrating block of mass
m
are made
to vibrate on Earth, and their respective natural frequencies are
measured. The two systems are then taken to the Moon and are
again allowed to vibrate at their respective natural frequencies. How
will the natural frequency of each system change when compared
with that on the Earth, and which of the two systems will experience
the larger change in natural frequency?
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Problem 9.6
The uniform disk of radius Rand thickness tis attached to the thin shaft of radius
r, length L, and negligible mass. The end Aof the shaft is fixed. From mechanics
of materials, it can be shown that if a torque
M´
is applied to the free end of the
shaft, then it can be related to the twist angle via
DM´L
GJ ;
where
G
is the shear modulus of elasticity of the shaft and
JD
2r4
is the polar
moment of inertia of the cross-sectional area of the shaft. Letting
be the mass
density of the disk. Using the given relationship between
M´
and
, determine
the natural frequency of vibration of the disk in terms of the given dimensions and
material properties when it is given a small angular displacement
in the plane of
the disk.
Kinematic Equations. To find the equation of motion, we have that ˛dDR
.
LD1
2R4tR
LR4tD0:
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Dynamics 2e 1929
Problem 9.7
A rigid body of mass
m
, mass center at
G
, and mass moment of inertia
IG
is pinned at
an arbitrary point Oand allowed to oscillate as a pendulum.
By writing the Newton-Euler equations, determine the distance
`
from
G
to the
pivot point
O
so that the pendulum has the highest possible natural frequency of
oscillation.
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permission of McGraw-Hill, is prohibited.
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 1931
Problem 9.8
A rigid body of mass
m
, mass center at
G
, and mass moment of inertia
IG
is pinned at
an arbitrary point Oand allowed to oscillate as a pendulum.
Using the energy method, determine the distance
`
from
G
to the pivot point
O
so
that the pendulum has the highest possible natural frequency of oscillation.
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permission of McGraw-Hill, is prohibited.
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.