Last time: Defined Angular Velocity and Angular Acceleration Vectors:

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Last time Defined Angular Velocity and Angular
Acceleration Vectors
And could then express the velocity of a point a
on a rotating (fixed axis) rigid body as
same as
and the acceleration as
Tangential Acceleration
Normal Acceleration
same as
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General Motion Rotation Translation
Motion of a Rigid body can be described
as TRANSLATION of any one point on the body (say
B) plus ROTATION of the body AROUND B.
Nothing special about B any point will do
because rotation (?,?) are properties of the
entire body
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Analyze General Motion
Just geometry algebra
Lets take B as the reference point on the
body Define ? with respect to fixed coordinates
X, Y
?
But A and B are on the same rigid body
Velocity at A (Velocity at B)
(Rotational Velocity of A around B)
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What about acceleration? Just another time
derivative.
?
Acceleration at A (Acceleration at B)
(Rotational Acceleration of A around B)
ONLY 3 INDEPENDENT QUANTITIES DESCRIBE MOTION
xB, yB, ?
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Summary on Kinematics Motion of Rigid Body can
be described by TRANSLATIONAL MOTION
of any one point ROTATIONAL MOTION
of entire body
?
Vector description
Component description
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Dynamics of Rigid Bodies
What is the equivalent of F ma ? How does the
angular velocity change ? What are momentum,
kinetic energy, angular moment of a rigid body
? - Rigid body has both translation and rotation
What is the rotational analog of mass? i.e.
what is resistance to angular acceleration? ?
Moment of Inertia
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Dynamics of Rigid Bodies
Forces are applied at different locations on the
body Forces friction, normal reaction, gravity,
other applied forces
Forces causes the body to both translate and
rotate
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Need to build on our understanding of the motion
of point objects.
MASS and CENTER of MASS
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For now, just one quantity Angular momentum
Sum of angular momenta of masses mi at ri moving
at Vi
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Sum of the moments around the center of mass is
proportional to the angular acceleration
Coefficient of proportionality is the Moment of
Inertia IG
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(Mass) Moment of Inertia
r project distance perpendicular to the
rotation axis passing through G
Geometrical property of the rigid body
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For uniform mass density (which we will assume
unless otherwise stated)
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Simple example
4 masses at various distances Treat masses as
point masses (radii ltlt R)
Smaller mass makes larger contribution because it
is twice as far away
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Solid Cube
Calculate the mass
Now calculate the moment of inertia I
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(Mass) Moment of Inertia
Moment of Inertia can usually be expressed (for
simple shapes) in the form
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Solid Cylinder
Use cylindrical coordinates
Moment of Inertia along axis of cylinder is then
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PARALLEL AXIS THEOREM
Suppose we have a rigid body that does not rotate
around a Center of Mass axis, but around some
other parallel axis? What is appropriate Moment
of Inertia?
Moment of inertia is still defined in the same
way
Reflects distribution of mass relative to the
axis of rotation
G ? Center of mass axis O ? Axis of rotation
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Moment of inertia of a cylinder about axis O
along edge
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COMPOSITE BODIES
Suppose we have a complicated structure, but
could approximate it by putting together some
simpler shapes. What is I ?
Moment of Inertia is just a sum, so break sum up
into parts
Combined with parallel axis theorem, we can write
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For bodies with holes, just add and subtract
hole

-
Body with hole
Body without hole
Body in shape of hole