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Key Concepts: Motion of a rigid body in a plane can be thought
of as motion of the center of
mass
coupled with motion about the center of mass. Euler's First Law governs translation of
the
center
of mass whereas Euler's Second Law governs rotation about the center of
mass. Sometimes,
to
simplify the analysis, it is convenient to express Euler's Second Law about
an arbitrary point, P, other than the center of mass, C,
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In a Nut Shell: Euler’s second law requires that
the sum of the moments of the external
forces
acting on the rigid body must equal the change in the angular momentum of
the rigid body.
In
general the angular momentum, HP,
of a rigid body about an arbitrary point, P moving
with
the body, is given by HP = IPzzω + rPC x m vP, where IPzz
is the moment of inertia of
mass
of the body about point P, ω
is its angular velocity, rPC is
the position vector from
the
center of mass, C, to the arbitrary point, P, m is the mass of the body,
and vP
is the velocity
of
point P in the fixed frame of reference.
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Consider
a rigid body of mass, m, located at its center of mass, C. Then the general expression for Euler’s
second Law with reference to arbitrary point, P, is
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Σ MP =
dHP/dt |
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Euler’s Second Law |
where Σ MP is the sum
of all the moments of the external forces acting on the rigid
body
about point P and dHp/dt is the rate of change of the angular
momentum
(in
the inertial frame) of the rigid body about point P.
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Now dHP/dt = rPC x m aP + IPzz α + ω
x (IPzzω) and for
planar kinetics with symmetrical
bodies where the products of inertia, Ixz
and Iyz are zero, then ω x (IPω) = 0.
The result for planar motion is:
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Σ MP =
(rPC x m aP)z
+ IPzz α |
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Euler’s Second Law- Arbitrary
Point P |
Another
form is:
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Σ MP =
(rPC x m aC)z
+ ICzz α |
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Euler’s Second Law-
Arbitrary Point P |
where
( )z refers to z-component
only, m is the mass of the body, aC is
the acceleration of the mass
center,
aP is the acceleration of point P, ICzz is the mass moment of inertia
about point C, IPzz is
the
mass moment of inertia about point P, and
α is the angular acceleration of the body
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Click
here for examples.
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