Lecture 14a

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Chapter 8 Rotational Motion
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• Topic of Chapter Objects rotating
• First, rotating, without translating.
• Then, rotating AND translating together.
• Assumption Rigid Body
• Definite shape. Does not deform or change shape.
• Rigid Body motion Translational motion of
  center of mass (everything done up to now)
  Rotational motion about an axis through center of
  mass. Can treat the two parts of motion
  separately.

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• COURSE THEME NEWTONS LAWS OF MOTION!
• Chs. 4 - 7 Methods to analyze the dynamics of
  objects in TRANSLATIONAL MOTION. Newtons Laws!
• Chs. 4 5 Newtons Laws using Forces
• Ch. 6 Newtons Laws using Energy Work
• Ch. 7 Newtons Laws using Momentum.
• NOW
• Ch. 8 Methods to analyze dynamics of objects in
  ROTATIONAL LANGUAGE. Newtons Laws in Rotational
  Language!
• First, Rotational Language. Analogues of each
  translational concept we already know!
• Then, Newtons Laws in Rotational Language.

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Rigid Body Rotation
A rigid body is an extended object whose size,
shape, distribution of mass dont change as the
object moves and rotates. Example a CD ?
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Three Basic Types of Rigid Body Motion
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Pure Rotational Motion
All points in the object move in circles about
the rotation axis (through the Center of Mass)
r
Reference Line
The axis of rotation is through O is ? to the
picture. All points move in circles about O
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In purely rotational motion, all points on the
object move in circles around the axis of
rotation (O). The radius of the circle is R.
All points on a straight line drawn through the
axis move through the same angle in the same
time.
r
r
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Sect. 8-1 Angular Quantities
r
Positive Rotation!

• Description of rotational
• motion Need concepts
• Angular Displacement
• Angular Velocity, Angular Acceleration
• Defined in direct analogy to linear quantities.
• Obey similar relationships!

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• Rigid object rotation
• Each point (P) moves
• in a circle with the
• same center!
• Look at OP When P
• (at radius R) travels an
• arc length l, OP sweeps
• out angle ?.
• ? ? Angular Displacement of the object

r
? Reference Line
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• ? ? Angular Displacement
• Commonly, measure ? in degrees.
• Math of rotation Easier if
• ? is measured in Radians
• 1 Radian ? Angle swept out
• when the arc length radius
• When ? ? R, ? ? 1 Radian
• ? in Radians is defined as
• ? ratio of 2 lengths (dimensionless)
• ? MUST be in radians for this to be valid!

r
? Reference
Line
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• ? in Radians for a circle of radius r, arc length
  ? is defined as ? ? (?/r)
• Conversion between radians degrees
• ? for a full circle 360º (?/r) radians
• Arc length ? for a full circle 2pr
• ? ? for a full circle 360º 2p radians
• Or 1 radian (rad) (360/2p)º ? 57.3º
• Or 1º (2p/360) rad ? 0.017 rad
• In doing problems in this chapter, put your
  calculators in RADIAN MODE!!!!

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Example 8-2

• ? ? 3?10-4 rad ? º
• r 100 m, ? ?
• a) ? (3?10-4 rad)
• ?(360/2p)º/rad 0.017º
• b) ? r? (100) ? (3?10-4)
• 0.03 m 3 cm
• ? MUST be in radians in part b!

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Angular Displacement
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Angular Velocity(Analogous to linear velocity!)

• Average Angular Velocity
• angular displacement ?? ?2 ?1
• (rad) divided by time ?t
• (Lower case Greek omega, NOT w!)
• Instantaneous Angular Velocity
• (Units rad/s) The SAME for all points
• in the object! Valid ONLY if ? is in rad!

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Angular Acceleration(Analogous to linear
acceleration!)

• Average Angular Acceleration change in angular
  velocity ?? ?2 ?1 divided by time ?t
• (Lower case Greek alpha!)
• Instantaneous Angular Acceleration limit of a
  as ?t, ?? ?0
• (Units rad/s2)
• The SAME for all points in body! Valid ONLY for ?
  in rad ? in rad/s!

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Relations of Angular Linear Quantities
Ch. 5 (circular motion) A mass moving in a
circle has a linear velocity v a linear
acceleration a. Weve just seen that it also
has an angular velocity an angular
acceleration.
??
??
r
? There MUST be relationships between the
linear the angular quantities!
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Connection Between Angular Linear Quantities
Radians! ?
v (??/?t), ?? r?? ? v r(??/?t) r?

• v r?
• ? Depends on r
• (? is the same for all points!)

vB rB?B, vA rA?A vB gt vA since rB gt rA
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Summary Every point on a rotating body has an
angular velocity ? and a linear velocity v. They
are related as
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Relation Between Angular Linear Acceleration

• In direction of motion
• (Tangential acceleration!)
• atan (?v/?t), ?v r??
• ? atan r (??/?t)
• atan ra
• atan depends on r
• a the same for all points

_____________
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Angular Linear Acceleration

• From Ch. 5 there is also
• an acceleration ? to the
• motion direction (radial or
• centripetal acceleration)
• aR (v2/r)
• But v r?
• ? aR r?2
• aR depends on r
• ? the same for all points

_____________
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Total Acceleration

• ? Two ? vector components
• of acceleration
• Tangential
• atan ra
• Radial
• aR r?2
• Total acceleration
• vector sum
• a aR atan

_____________
a ?---
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Relation Between Angular Velocity Rotation
Frequency

• Rotation frequency
• f revolutions / second (rev/s)
• 1 rev 2p rad
• ? f (?/2p) or ? 2p f angular frequency
• 1 rev/s ? 1 Hz (Hertz)
• Period Time for one revolution.
• ? T (1/f) (2p/?)

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Translational-Rotational Analogues Connections

• ANALOGUES
• Translation Rotation
• Displacement x ?
• Velocity v ?
• Acceleration a a
• CONNECTIONS
• ? r?, v r?
• atan r a
• aR (v2/r) ?2 r

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Correspondence between Linear Rotational
quantities
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Conceptual Example 8-3 Is the lion
faster than the horse?
On a rotating merry-go-round, one child sits on a
horse near the outer edge another child sits on
a lion halfway out from the center. a. Which
child has the greater translational velocity v?
b. Which child has the greater angular velocity
??
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Example 8-4 Angular Linear Velocities
Accelerations
A merry-go-round is initially at rest (?0
0). At t 0 it is given a constant angular
acceleration a 0.06 rad/s2. At t 8 s,
calculate the following a. The
angular velocity ?.
b. The linear velocity v of a child located r
2.5 m from the center.
c. The tangential (linear) acceleration atan of
that child. d. The centripetal acceleration aR of
the child. e. The total linear acceleration a of
the child.
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Example 8-5 Hard Drive
The platter of the hard drive of a computer
rotates at frequency f 7200 rpm (rpm
revolutions per minute rev/min) a. Calculate
the angular velocity ? (rad/s) of the platter. b.
The reading head of the drive r 3 cm ( 0.03 m)
from the rotation axis. Calculate the linear
speed v of the point on the platter just below
it. c. If a single bit requires 0.5 µm of length
along the direction of motion, how many bits per
second can the writing head write when it is r
3 cm from the axis?