Dynamics 2e 1933
Problem 9.9
A block of mass
mD3kg
is in equilibrium when a hammer hits it,
imparting a velocity
v0
of
2m=s
to it. If
k
is
120 N=m
, determine
the amplitude of the ensuing vibration and find the maximum
acceleration experienced by the block.
mxD0:
The initial velocity of the mass is
Px.0/ Dv0D2m=s
and its initial position is
x.0/ Dx0D0
, therefore,
since !2
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Problem 9.10
A construction worker
C
is standing at the midpoint of a
14 ft
long pine board that is simply supported. The board is a standard
212
, so its cross-sectional dimensions are as shown. Assuming the
worker weighs
180 lb
and he flexes his knees once to get the board
oscillating, determine his vibration frequency. Neglect the weight
of the beam, and use the fact that a load
P
applied to a simply
supported beam will deflect the center of the beam
PL3=.48EIcs/
,
where
L
is the length of the beam,
E
is its modulus of elasticity, and
Ics
is the area moment of inertia of the cross section of the beam.
The elastic modulus of pine is 1:8 106psi.
dt .T CV / Dconstant;where TD1
Force Laws.
The force due to the board is assumed to be linear
elastic with the force law
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Dynamics 2e 1935
Using the fact that
mg Dkeqıst
and substituting in the expression for the spring constant of the board, this
equation becomes the standard harmonic oscillator equation
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Problem 9.11
A truck drives onto a deck scale to be weighed, thus causing the
truck and scale to vibrate vertically at the natural frequency of the
system. The empty truck weighs
74;000 lb
, the scale platform weighs
51;000 lb
, and the platform is supported by eight identical springs
(four of which are shown), each with constant
kD3:6 105lb=ft
.
Modeling the truck, its contents, and the concrete deck as a single
particle, if a vibration frequency of
3:3 Hz
is measured, what is the
weight of the payload being carried by the truck?
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Dynamics 2e 1937
Problem 9.12
The buoy in the photograph can be modeled as a circular cylinder
of diameter
d
and mass
m
. If the buoy is pushed down in the
water, which has density
, it will oscillate vertically. Determine
the frequency of oscillation. Evaluate your result for
dD1:2
m,
mD900 kg
, and surface seawater, which has a density of
D
1027 kg=m3
.Hint: Use Archimedes’ principle, which states that
a body wholly or partially submerged in a fluid is buoyed up by a
force equal to the weight of the displaced fluid.
22
where ˝is the volume of the fluid displaced.
Kinematic Equations. Since we want the equation of motion, we let ayD Ry.
Computation. Substituting the force law and the kinematic equation into Eq. (1), it becomes
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Problem 9.13
The L-shaped bar lies in the vertical plane and is pinned at
O
. One end
of the bar has a linear elastic spring with constant
k
attached to it, and
attached at the other end is a sphere of mass
m
and negligible size. The
angle
is measured from the equilibrium position of the system, and it
is assumed to be small.
Assuming that the L-shaped bar has negligible mass, determine the
natural period of vibration of the system by writing the Newton-Euler
equations.
md 2Dh
drk
m)D2d
hrm
k.
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permission of McGraw-Hill, is prohibited.
Dynamics 2e 1939
Problem 9.14
The L-shaped bar lies in the vertical plane and is pinned at
O
. One end
of the bar has a linear elastic spring with constant
k
attached to it, and
attached at the other end is a sphere of mass
m
and negligible size. The
angle
is measured from the equilibrium position of the system, and it
is assumed to be small.
Assuming that the L-shaped bar has negligible mass, determine the
natural period of vibration of the system using the energy method.
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Problem 9.15
The L-shaped bar lies in the vertical plane and is pinned at
O
. One end
of the bar has a linear elastic spring with constant
k
attached to it, and
attached at the other end is a sphere of mass
m
and negligible size. The
angle
is measured from the equilibrium position of the system, and it
is assumed to be small.
Assuming that the L-shaped bar has mass per unit length
, determine
the natural period of vibration of the system by writing the Newton-Euler
equations.
Solution
Including the mass of the L-shaped bar, the FBD of the system is as shown
on the right.
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Dynamics 2e 1941
Problem 9.16
The L-shaped bar lies in the vertical plane and is pinned at
O
. One end
of the bar has a linear elastic spring with constant
k
attached to it, and
attached at the other end is a sphere of mass
m
and negligible size. The
angle
is measured from the equilibrium position of the system, and it
is assumed to be small.
Assuming that the L-shaped bar has mass per unit length
, determine
the natural period of vibration of the system using the energy method.
Solution
Including the mass of the L-shaped bar, the FBD of the system is as
shown on the right. From this FBD we can see that the spring force
Fs
and the weight forces
mg
and
gd
do work and thus the system is
conservative.
Balance Principles. Applying the energy method, which is
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permission of McGraw-Hill, is prohibited.
or, canceling P
in every term, this becomes
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