Rotational Kinematics

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Rotational Kinematics
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Circular Motion
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A Particle inUniform Circular Motion
For a particle in uniform circular motion, the
velocity vector v remains constant in magnitude,
but it continuously changes its direction.
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Angular Position ?
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Angular Position q
Degrees and revolutions
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Angular Position q
Arc length s, measured in radians
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Angular Velocity w
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Sign of ?
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Connections BetweenLinear Rotational Quantities
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Angular Acceleration a
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Comparison to 1-D Kinematics

• Angular Linear

And for a point at a distance R from the rotation
axis
x R????????????v ?R ??????????a ?R
By convention, ?, ?, ? are positive if they are
in the counterclockwise direction.
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Decelerating Windmill
As the wind dies, a windmill that had been
rotating at w 2.1 rad/s begins to slow down at
a constant angular acceleration of a -0.45
rad/s2. How long does it take for the windmill to
come to a complete stop?
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Angular Velocity Acceleration ACT
The fan blade shown is slowing down. Which
option describes a and w?
(c) wlt0 and agt0
(d) wlt0 and alt0.
(a) wgt0 and agt0
(b) wgt0 and alt0
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Rotational Kinematics
If the angular acceleration is constant
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Thrown for a Curve
To throw a curve ball, a pitchergives the ball
an initial angularspeed of 157.0 rad/s. When
thecatcher gloves the ball 0.795 slater, its
angular speed hasdecreased (due to air
resistance)to 154.7 rad/s. (a) What is the
balls angular acceleration, assuming it to be
constant? (b) How many revolutions does the ball
make before being caught?
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Wheel of Misfortune
On a certain game show, contestants spin
thewheel when it is their turn. One contestant
givesthe wheel an initial angular speed of 3.40
rad/s.It then rotates through 1.25 revolutions
andcomes to rest on BANKRUPT. (a) Find the
wheels angular acceleration,assuming it to be
constant. (b) How long does it take for the
wheel tocome to rest?
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A Rotating Crankshaft
A cars tachometer indicates the angular
velocity w of the crank shaft in rpm. A car
stopped at a traffic light has its engine idling
at 500 rpm. When the light turns green, the
crankshafts angular velocity speeds up at a
constant rate to 2500 rpm in a time interval of
3.0 s. How many revolutions does the crankshaft
make in this time interval?
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Time to Rest

• A pulley rotating in the counterclockwise
  direction is attached to a mass suspended from a
  string. The mass causes the pulleys angular
  velocity to decrease with a constant angular
  acceleration a -2.10 rad/s2.
• (a) If the pulleys initial angular velocity is
  w0 5.40 rad/s, how long does it take for the
  pulley to come to rest?
• Through what angle does the pulley turn during
  this time?

(c) If the radius of the pulley is 5.0 cm,
through what distance is the mass lifted?
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CD Speed
CDs and DVDs turn with a variable w that keeps
the tangential speed vt constant. Find the
angular speed w and the frequency that a CD must
have in order to give it a linear speed vt
1.25 m/s when the laser beam shines on the
disk (a) at 2.50 cm from its center, and(b) at
6.00 cm from its center.
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Rotational vs. Linear Kinematics
Analogies between linear and rotational
kinematics
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Connections BetweenLinear Rotational Quantities
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More Connections BetweenLinear Rotational
Quantities
This merry-go-round has both tangential and
centripetal acceleration.
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The Microhematocrit

• Suppose the centrifuge is just starting up, and
  that it has an angular speed of 8.00 rad/s and an
  angular acceleration of 95.0 rad/s2.
• (a) What is the magnitude of the centripetal,
  tangential, and total acceleration of the bottom
  of a tube?
• (b) What angle does the total acceleration make
  with the direction of motion?