Rotation of Rigid Bodies

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Rotation of Rigid Bodies
Rotational Motion in close analogy with linear
motion (distance/displacement, velocity,
acceleration) Angular measure in natural
units Angles and Rotation in radians
r
Angle arc length / radius
s
q
from one complete circuit, 360o 2p rad 45o
p/4 rad 90o p/2 rad 180o p rad 1 rad 57.30o
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Angular velocity an object which rotates about a
fixed axis has an average angular velocity wav
usually rad/s but sometime rpm,
rps instantaneous angular velocity is given by
s
r
q
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Angular Acceleration the rate of change of
angular speed
acw2r
aTar
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Example the angular position of a flywheel is
given by q (2.00 rad/s3) t3. The diameter of
this flywheel is .360 m. Find the angular
displacement at 2.00s and at 5.00s. Find the
average angular velocity between 2.00s and
5.00s. Find the instantaneous angular velocity at
2.00s, 3.50s and 5.00s. Find the average angular
acceleration between 2.00s and 5.00s. Find the
instantaneous angular acceleration at
3.50s. Calculate the speed of a point on the edge
of the flywheel at 3.50s. Calculate the
tangential and radial acceleration of a point on
the edge of the flywheel at 3.50s.
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Rotation with constant angular acceleration (just
like linear 1-d)
watch units!!!
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Example A wheel with an initial angular velocity
of 4.00 rad/s undergoes a constant acceleration
of -1.20 rad/s2. What is the angular
displacement and angular velocity at t
3.00s? How many rotations does the wheel make
before coming to rest?
Example A discus thrower turns with an angular
acceleration of 50 rad/s2, moving the discus
around a constant radius of .800 m. Find the
tangential and centripetal acceleration when the
discus has an angular velocity of 10 rad/s.
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Example An airplane propeller is to rotate at
maximum of 2400 rpm while the aircrafts forward
velocity is 75.0 m/s. How big can the propeller
be if the the speed of the tips relative to the
air is not to exceed 270 m/s? At this speed,
what is the acceleration of the propeller tip?
Example discuss chain-linked gears, belt drives
etc linear velocity vs angular velocity.
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v
Rotational Kinetic Energy for a single point
particle
m
r
m1
v1
v3
for a solid rotating object, piece by piece
r1
r2
m2
r3
m3
v2
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Example Three masses are connected by light
bracing as shown. What is the moment of inertia
about each of the axis shown? What would the
kinetic energy for be for rotation at 4.00 rad/s
about each of the axis shown?
.10kg
.50m
.30m
.20kg
.40m
.30kg
axis perpendicular to plane
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Moments of inertia for some common geometric
solids
a
b
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A cord is wrapped around a solid 50 kg cylinder
which has a diameter of 0.120 m, and which
rotates (frictionlessly) about an axis through
its center. A 9.00 N force is applied to the end
of the cable, causing the cable to unwind and the
drum (initially at rest) to rotate. After the
cable has unwound a distance 2.00m, determine the
work done by the force, the kinetic energy of the
drum, the rotational velocity of the drum, and
the speed of the unwinding cable.
9.00 N
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Combining Translation and Rotation KE
KEtranslation KErotation ½mv2 ½Iw2
A connection for rolling without slipping s q
r v w r a a r, a angular
acceleration Gravitational Potential Energy for
an extended object use center of mass U
mgycm The Great Race 2 objects rolling (from
rest) down the same incline lost PE gained
KE same radius, object with the smallest I has
most v gt wins race
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A mass m is suspended by a string wrapped around
a pulley of radius R and moment of inertia I.
The mass and pulley are initially at rest. After
the mass has dropped a height h, determine the
relation between the final speed of the mass and
the given parameters (m, I, h). Examine the
special case where the pulley is a uniform disk
of mass M.
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The Parallel Axis Theorem The moment of Inertia
about an axis is related in a simple way to the
moment of inertia about a parallel axis which
runs through the center of mass Ip Icm MR2
p
a
b
cm
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Example An 3.6 kg object is found to have a
moment of inertia of .132 kg m2 about an axis
which is found to be .15m from is center of mass.
What is the moment of inertia of this object
about a parallel axis which does go through the
objects center of mass?
Example Find the moment of inertia of a thin
uniform disk about an axis perpendicular to its
flat surface, running along the edge of the disk.
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Beyond discrete masses adventures in vector
calculus!