ENGR 220 Lecture 12: Rigid Body Rotation and Translation

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ENGR 220 Lecture 12 Rigid Body Rotation and
Translation
Two- dimensional Distance between points on body
is fixed
Planar Rigid Body Motion

• Translation
• body moves on a plane with no rotation
• All points on body have same velocity vector
• Rotation
• Body center of mass doesnt move, but all points
  rotate in circular motion
• Axis of rotation is perpendicular to plane and is
  the same for all points
• Any single point remains in same plane during
  motion
• General
• Translation and rotation both occur at once on 2D
  reference plane

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Translation Motion only take points A and B on
a body
Position
Diagram
Velocity (derivative of position)
Acceleration (derivative of velocity)
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Rigid Body Translation Equations of Motion
or
Note could also use -
Rectilinear motion
Curvilinear motion
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Rotational Motion Only
Rotation ? circular paths for all
particles on body ? points on body still have
position, velocity, acceleration ? kinematic
equations still hold, but with angular
variables ? use polar coordinates (see diagram)
Diagrams
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Rotational Motion Analogs to Translation
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Rigid Body Rotation Equation of Motion
M represents resistance to change in translation,
or acceleration
Rotation
I represents resistance to change in rotation, or
angular acceleration
Transform mass integral into spatial integral
Moment of Inertia
Density comes out of integral if it is constant
throughout object
Parallel Axis theorem Use to find I about an
axis that is not at the center of gravity of the
body
Radius of gyration
Total moment of inertia about axis of interest
Moment of inertia about c.o.g.
Mass of object times distance between axis of
interest and c.o.g. squared
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Example Ch 7.1 6 Two cases. 1500lb dragster
to travel 200ft operating at limit of no-slip
condition with ?d0.82. Case 1 rides with front
wheels up. Case 2 rides with 85 of weight on
rear tires.
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Example Ch 7.2 13 45/45/90 triangle with areal
density 4kg/m2 and short side length 0.5m
connected to semicircle with areal density
10kg/m2. Find IO for in-plane rotation.
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Example Ch 7.2 15 Solid of revolution formed
by rotating zax2 about z-axis and filling in
shell. Find mass moment of inertia about z axis
for shell going from z0 to zz0. Express in
terms of mass m and again in terms of density ?.
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Example Ch 7.2 34 Four balls at ends of light
rods of half-length e are centered on cylinder of
radius r fixed at O. Rope pf length L is pulled
down with force F to cause ccw rotation. a. Find
time to pull full length of rope b. What is
change in time of r is doubled? c. Compare final
angular speed for each case