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Dynamics 2e 1963
Problem 9.28
The U-tube manometer lies in the vertical plane and contains a fluid of density
that has been displaced a distance
y
and oscillates in the tube. If the cross-
sectional area of the tube is
A
and the total length of the fluid in the tube is
L
,
determine the natural period of oscillation of the fluid, using the energy method.
Hint: As long as the curved portion of the tube is always filled with liquid (i.e.,
the oscillations do not get large enough to empty part of it), the contribution of
the liquid in the curved portion to the potential energy is constant.
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.
Computation.
Substituting in the kinematics equation,
T
, and
V
and applying the energy method, we
obtain
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 1965
Problem 9.29
The uniform semicylinder of radius
R
and mass
m
rolls without slip on
the horizontal surface. Using the energy method, determine the period of
oscillation for small .
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.
Computation.
Substituting Eq. (2) into Eq. (1) and then using Eqs. (3) and (1) to apply the energy method
to the semicylinder, we obtain the following equation of motion:
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 1967
Problem 9.30
The thin-shell semicylinder of radius
R
and mass
m
rolls without slip on
the horizontal surface. Using the energy method, determine the period of
oscillation for small .
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.
We can now find IGusing the parallel axis theorem as
Finally, we can find IOby again using the parallel axis theorem as
where we have approximated `by Rdfor small .
where we have used the small angle approximation for cos .
Kinematic Equations. Since we want the equation of motion, we let !sDP
.
Computation.
Substituting
IO
and the kinematic equation into Eq. (1) and then using Eqs. (2) and (1) to
apply the energy method to the thin shell semicylinder, we obtain the following equation of motion:
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 1969
Problem 9.31
The magnification factor for a forced (undamped) harmonic oscillator is measured to be equal to
5
.
Determine the driving frequency of the forcing if the natural frequency of the system is 100 rad=s.
Solution
The
MF
for a harmonic oscillator depends on the forcing frequency and the natural frequency of a system
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.
