Physics 221 Chapter 10

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Physics 221Chapter 10
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Problem 1 . . . Angelas new bike

• The radius of the wheel is 30 cm and the speed v
  5 m/s. What is the rpm (revolutions per minute)
  ?

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Solution 1 . . . Angelas rpm

• r radius
• circumference 2 ? r
• f revolutions per second
• v d/t
• v 2 ? f r
• 5 (2 ?)(f)(0.3)
• f 2.6 revolutions per second or 159 rpm

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What is a Radian?

• A radian is about 60 degrees which is 1/6 of
  the circle (360 degrees).
• To be EXACT, the radian pie has an arc equal to
  the radius.

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Problem 2 What EXACTLY is a Radian?

• A. 550
• B. 570
• C. 590
• D. 610

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Solution 2 What EXACTLY is a Radian?

• If each pie has an arc of r, then there must be
  2 ? radians in a 3600 circle.
• 2 ? radians 3600
• 6.28 radians 3600
• 1 radian 57.30

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Angular Velocity

• Angular Velocity radians / time
• ? ? / t

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Problem 3 . . . Angular Velocity

• The radius of the wheel is 30 cm. and the
  (linear) velocity, v, is 5 m/s. What is the
  angular velocity?

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Solution 3 . . . Angular Velocity

• We know from problem 1 that
• f 2.6 rev/s
• But 1 rev 2 ? radians
• So ? ? / t
• ? (2.6)(2 ?) /(1 s)
• ? 16.3 rad/s

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V and ?

• Linear (m/s)
  Angular (rad/s)
• V
  ?
• d / t
  ? / t
• 2 ? r f / t
  2 ? f / t
• v r ?

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a and ?

• Linear (m/s2)
  Angular (rad/s2)
• a
  ?
• ( Vf - Vi ) / t
  ( ?f - ?i ) / t
• a r ?

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Problem 4 . . .Your CD player

• A 120 mm CD spins up at a uniform rate from rest
  to 530 rpm in 3 seconds. Calculate its
• (a) angular acceleration
• (b) linear acceleration

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Solution 4 . . . CD player

• ? ( ?f - ?i ) / t
• ? (530 x 2 ? /60 - 0) / 3
• ? 18.5 rad/s2
• a r ?
• a 0.06 x 18.5
• a 1.1 m/s2

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Problem 5 . . . CD Music

• To make the music play at a uniform rate, it is
  necessary to spin the CD at a constant linear
  velocity (CLV). Compared to the angular velocity
  of the CD when playing a song on the inner track,
  the angular velocity when playing a song on the
  outer track is
• A. more
• B. less
• C. same

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Solution 5 . . . CD Music

• v r ?
• When r increases, ? must decrease in order for v
  to stay constant. Correct answer B
• Note Think of track races. Runners on the
  outside track travel a greater distance for the
  same number of revolutions!

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Angular Analogs

• d
  ?
• v
  ?
• a
  ?

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Problem 6 . . . Angular Analogs

• d Vi t 1/2 a t2
  ?

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Solution 6 . . . Angular Analogs

• d Vi t 1/2 a t2
  ? ?i t 1/2 ? t2

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Problem 7 . . . Red Corvette

• The tires of a car make 65 revolutions as the car
  reduces its speed uniformly from 100 km/h to 50
  km/h. The tires have a diameter of 0.8 m. At
  this rate, how much more time is required for it
  to stop?

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Solution 7 . . . Corvette

• 100 km/h 27.8 m/s 69.5 rad/s since v r ?
• Similarly 50 km/h 34.8 rad/s
• (?f)2 (?i)2 2 ? ?
• (34.8)2 (69.5)2 (2)(?)(65)(6.28)
• ? - 4.4 rad/s2
• ?f ?i ? t
• 0 34.8 - 4.4 t
• t 7.9 s

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Torque

• Torque means the turning effect of a force.
• SAME force applied to both. Which one will turn
  easier?

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Torque

• Torque distance x force
• ? r x F

Easy!
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Torque

• Which one is easier to turn?

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Torque . . . The Rest of the Story!

• ? r F sin ?

Easy!
?
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Problem 8 . . . Inertia Experiment

• SAME force applied to m and M. Which one
  accelerates more?

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Solution 8 . . . Inertia Experiment

• Since F ma, the smaller mass will accelerate
  more

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Problem 9 Moment of Inertia Experiment

• SAME force applied to all. Which one will
  undergo the greatest angular acceleration?

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Solution 9 Moment of Inertia Experiment

• This one will undergo the greatest angular
  acceleration.

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What is Moment of Inertia?

• F m a
• Force mass x ( linear ) acceleration
• ? I ?
• Torque moment of inertia x angular
  acceleration

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I mr2

• The moment of inertia of a particle of mass m
  spinning at a distance r is I mr2
• For the same torque, the smaller the moment of
  inertia, the greater the angular acceleration.

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All about Sarah Hughes . . .

• Click me!

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Problem 10 . . . Sarah Hughes

• Will her mass change when she pulls her arms in?
• Will her moment of inertia change?

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Solution 10 . . . Sarah Hughes

• Mass does not change when she pulls her arms in
  but her moment of inertia decreases.

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Problem 11 . . . Guessing Game

• A ball, hoop, and disc have the same mass.
  Arrange in order of decreasing I
• A. hoop, disc, ball
• B. hoop, ball, disc
• C. ball, disc, hoop
• D. disc, hoop, ball

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Solution 11 . . . Guessing Game

• I (moment of inertia) depends on the distribution
  of mass. The farther the mass is from the axis
  of rotation, the greater is the moment of
  inertia.
• I MR2 I 1/2 MR2
  I 2 /5 MR2
• hoop disc
  ball

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Problem 12 . . . K.E. of Rotation

• What is the formula for the kinetic energy of
  rotation?
• A. 1/2 mv2
• B. 1/2 m?2
• C. 1/2 I?2
• D. I ?

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Solution 12 . . . K.E. of Rotation

• The analog of v is ?
• The analog of m is I
• The K.E. of rotation is 1/2 I?2

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Problem 13 . . . Long, thin rod

• Calculate the moment of inertia of a long thin
  rod of mass M and length L rotating about an axis
  perpendicular to the length and located at one
  end.

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Solution 13 . . . Long, thin rod

• I mr 2
• However, r is a variable so we need to integrate.
  (aint that fun!)
• A small mass m of length dr must M/L dr
• I M/L ? r2 dr
• I (M/L)(L3 / 3 )
• I 1/3 ML2

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Problem 14 . . . In the middle

• Suppose the rod spins about its C.M. One can use
  the Parallel Axis Theorem to calculate ICM
• ID ICM MD2
• D is the distance between the C.M. and the other
  axis of rotation

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Solution 14 . . . In the middle

• ID ICM MD2
• 1/3 ML2 ICM M(L/2)2
• ICM 1/3 ML2 - 1/4 ML2
• ICM 1/12 ML2

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Problem 1 The race of the century!

• Will it be the hoop or the disc?

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Solution 1 . . . Race of the Century Hoop Loses
! ! !

• P.E. K.E. (linear) K.E. (angular)
• mgh 1/2 mv2 1/2 I?2
• mgh 1/2 mv2 1/2 I (v/r)2
• For the disc, I 1/2 mr2
• So mgh 1/2 mv2 1/2 (1/2 mr2)(v/r)2
• Disc v (4/3 g h)1/2
• Similarly Hoop v (g h)1/2