Dynamics 2e 1953
Problem 9.22
Grandfather clocks keep time by advancing the hands a set amount per oscillation of the pendulum.
Therefore, the pendulum needs to have a very accurate period for the clock to keep time accurately. As a
fine adjustment of the pendulum’s period, many grandfather clocks have an adjustment nut on a bolt at
the bottom of the pendulum disk. Screwing this nut inward or outward changes the mass distribution of
the pendulum by moving the pendulum disk closer to or farther from the axis of rotation at
O
. Model the
pendulum as a uniform disk of radius
r
and mass
mp
at the end of a rod of negligible mass and length
Lr, and assume that the oscillations of are small. Let mpD0:7 kg and rD0:1 m.
The clock is running slow so that it is losing 2 minutes every 24 hours (i.e., the clock takes 1442
minutes to complete a 1440-minute day). If the pendulum disk is at
LD0:85
m, how many turns of the
adjustment nut would be needed, and in what direction, to correct the pendulum’s period if the screw lead
is 0:5 mm?
Solution
The FBD at the right reflects the fact that we are neglecting the weight of the
adjustment nut and the rod connecting the disk to point O.
Balance Principles.
We can see from the FBD that, as the pendulum swings,
the weight of the pendulum disk is the only force that does work. Since the
system is conservative, we can apply the energy method, which is
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Kinematic Equations. Since we need to find the equation of motion, we let !pDP
.
Computation. Substituting the kinetic and potential energies into the energy method, we obtain
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Dynamics 2e 1955
Problem 9.23
The uniform cylinder of mass mand radius Rrolls without slipping on the
inclined surface. The spring with constant
k
wraps around the cylinder as it
rolls.
Determine the equation of motion for the cylinder by writing its Newton-
Euler equations. Determine the numerical value of the period of oscillation
of the cylinder using kD30 N=m, mD10 kg, RD30 cm, and D20ı.
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permission of McGraw-Hill, is prohibited.
From Eq. (2), we can see that the natural frequency and period are given by
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Dynamics 2e 1957
Problem 9.24
The uniform cylinder of mass mand radius Rrolls without slipping on the
inclined surface. The spring with constant
k
wraps around the cylinder as it
rolls.
Determine the equation of motion for the cylinder using the energy
method. Determine the numerical value of the period of oscillation of the
cylinder using kD30 N=m, mD10 kg, RD30 cm, and D20ı.
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permission of McGraw-Hill, is prohibited.
1958 Solutions Manual
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permission of McGraw-Hill, is prohibited.
Dynamics 2e 1959
Problem 9.25
A uniform bar of mass
m
is placed off-center on two counter-rotating
drums
A
and
B
. Each drum is driven with constant angular speed
!0
,
and the coefficient of kinetic friction between the drums and the bar
is
k
. Determine the natural frequency of oscillation of the bar on
the rollers. Hint: Measure the horizontal position of
G
relative to the
midpoint between the two drums, and assume that the drums rotate
sufficiently fast so that the drums are always slipping relative to the
bar.
2xGNAh
2CxGDIG˛bar:(3)
Force Laws.
Since the counter-rotating drums must be slipping relative to the bar, the force laws are given
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permission of McGraw-Hill, is prohibited.
Therefore, the natural frequency of oscillation is
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Dynamics 2e 1961
Problem 9.26
The uniform cylinder
A
of radius
r
and mass
m
is released from
a small angle
inside the large cylinder of radius
R
. Assuming
that it rolls without slipping, determine the natural frequency and
period of oscillation of A.
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permission of McGraw-Hill, is prohibited.
Problem 9.27
The uniform sphere
A
of radius
r
and mass
m
is released from
a small angle
inside the large cylinder of radius
R
. Assuming
that it rolls without slipping, determine the natural frequency and
period of oscillation of the sphere.
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.