A jet ski expels water at a rate of 1440. liters per minute at a velocity of 45.00 m/s. What thrust does it produce?

1
A jet ski expels water at a rate of 1440. liters
per minute at a velocity of 45.00 m/s. What
thrust does it produce?
Physics 1710Warm-up Quiz

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1. 45.00 N
2. 1440. N
3. 1080. N
4. 14112. N

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Physics 1710Chapter 10 Rotating Bodies
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• Solution

F jet ski - F water jet F water jet dp/dt
d(mv)/dt F water jet v dm/dt (45 m/s)(1440/60
kg/s) (45 m/s)(80 kg/s) 1080 N
How does a canoe paddle work?
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Physics 1710Chapter 10 Rotating Bodies
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• 1' Lecture
• The motion of a system of point particles is a
  combination of motion of the center of mass (CM)
  and the motion about the CM.
• Angular displacement is angle through which a
  body has rotated.
• Instantaneous angular speed is the time rate of
  angular displacement.
• Instantaneous angular acceleration is the time
  rate of change in angular speed.

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Physics 1710Chapter 10 Rotating Bodies
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• Center of Mass (CM)
• RCM ?mi ri / ?mi M ?mi

m1
m2
r1
r2
RCM
m3
r3
RCM is the mass-weighted mean position.
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Physics 1710Chapter 10 Rotating Bodies
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• Center of Mass (CM)
• RCM ? rdm / M
• M ? dm

RCM
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Physics 1710Chapter 10 Rotating Bodies
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• Total Linear Momentum
• RCM ?mi ri / M
• vCM d RCM /dt (1/M) ?i mi d ri /dt
• vCM (1/M) ?i mi vi
• Thus
• PCM ?i pi total p
• aCM d vCM /dt (1/M) ?i mi d vi /dt
• aCM (1/M) ?i mi d vi /dt
• Thus
• FCM M aCM ?i mi ai

? Force acts as if mass all mass were at CM.
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Physics 1710Chapter 10 Rotating Bodies
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• The center of mass (CM) of a system of particles
  of combined mass M moves like an equivalent
  particle of mass M would move under the
  influence of the resultant external force on the
  system.

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Physics 1710Chapter 10 Rotating Bodies
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Physics 1710Chapter 10 Rotating Bodies
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Angular Rotation s r ? ? s/r One radian is
the angle subtended by an arc length equal to the
radius of the arc. 1 radian 180/p 57.32
10
Hold your hand at arms length. What angle (in
radians and degrees) does it subtend?
Physics 1710Chapter 10 Rotating Bodies

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1. 0.01 rad, 1o.
2. 0.1 rad, 5o.
3. 0.2 rad, 10o
4. 1. rad, 60o.

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Physics 1710Chapter 10 Rotating Bodies
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• Solution

T s/R 4?/24? .17 rad 0.2 radian 10 o
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Physics 1710Chapter 10 Rotating Bodies
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• Average ?ave and Instantaneous Velocity ?
• Angular Displacement ?? ?f - ?I
• ?ave ?? / ?t
• ? lim?t ?0 ?? / ?t d?/dt
• ? is pronounced omega.

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Physics 1710Chapter 10 Rotating Bodies
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• Average and
• Instantaneous (Angular) Acceleration
• ?ave ?? / ?t
• ? lim?t ?0 ?? / ?t d? / dt
• ? is pronounced alpha.

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Physics 1710Chapter 10 Rotating Bodies
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• Rotational Kinematics
• (constant ?)
• ?f ?i ?t
• ?f ?i ?i t ½ ?t 2
• ?f 2 ?I 2 2?(?f -?i )
• v r ?
• a r ?
• a v 2 /r r ? 2

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Physics 1710Chapter 10 Rotating Bodies
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• Moment of Inertia
• I lim?mi?0 ?i ri 2 ? mi
• I ?r 2 dm
• I ??r 2 d V
• N.B. The moment of inertia is the second moment
  of the mass distribution, while the center of
  mass is the first moment.

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Physics 1710Chapter 10 Rotating Bodies
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• Moment of Inertia
• I ?i xi 2 ? mi

A
B
Which has greater moment of inertia about the
vertical axis? A or B?
17
Which has greater moment of inertia about the
vertical axis? A or B?
Physics 1710Chapter 10 Rotating Bodies

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3. A and B arr equal.

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Physics 1710Chapter 10 Rotating Bodies
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• Volume Integrals
• Cartesian Coordinates
• V ?dx dy dz x y z
• Cylindrical Coordinates
• V ? r dr dz d?
• (½ r 2) h (2?) ? r 2 h
• Spherical Coordinates
• V ? r 2 sin ? dr d? d?
• (? r 3)(2?) cos (0) cos (?) 4/3 ? r 3

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Physics 1710Chapter 10 Rotating Bodies
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• Parallel Axis Theorem
• The moment of inertia through a point, not the
  center of mass, is equal to the moment of
  inertia of the Center of Mass about the axis of
  rotation (MD 2) and the moment of inertia of the
  body about a parallel axis that passes through
  the center of mass
• I ICM MD 2

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Physics 1710Chapter 10 Rotating Bodies
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• Torque
• T r ? F
• Ti rj Fk rk Fj
• Tx ry Fz rz Fy
• Ty rz Fx rx Fz
• Tz rx Fy ry Fx

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Physics 1710Chapter 10 Rotating Bodies
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• Torque
• T r F sin ?
• In the direction ? to r ? to F.
• Use right hand rule.

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Physics 1710Chapter 10 Rotating Bodies
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• Torque Ladder Puzzle
• The torques are balanced. If one moves the mass
  higher on the ladder, should one move the weigh
  closer, farther or the same distance from the
  spar?

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Physics 1710Chapter 10 Rotating Bodies
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• Newtons Second Law for Rotation
• T I d?/dt I ?

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Physics 1710Chapter 10 Rotating Bodies
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• The Ring Race
• What will be the outcome in a race between a disk
  and a ring of equal mass and diameter? Will the
  ring win, lose or draw? Why?

25
Physics 1710Chapter 10 Rotating Bodies
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• Work and Energy
• in Rotational Motion
• T I ? I (d?/dt) I (d?/d?) (d?/dt)
• ?T d? ?I ? (d?/d?) d?
• Thus
• W ?I ? d?
• ½ I ?f 2 - ½ I ?i 2

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Physics 1710Chapter 10 Rotating Bodies
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• Work Rotational Energy Relation
• The net work done by external torques in rotating
  a symmetric rigid object about a fixed axis
  equals the change in the objects rotational
  energy.
• ?W ½ I ?f 2 - ½ I ?i 2

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Physics 1710Chapter 10 Rotating Bodies
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• Summary
• Angular displacement is the angle through which a
  body has rotated.
• Instantaneous angular speed is the time rate of
  angular displacement.
• Instantaneous angular acceleration is the time
  rate of change in angular speed.

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Physics 1710Chapter 10 Rotating Bodies
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• Summary (contd)
• The moment of inertia is the measure of the
  (inertial) resistance to angular acceleration and
  equal to the second moment of the mass
  distribution.
• Torque (twist) is the vector product of a
  force and the moment arm.

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Physics 1710Chapter 10 Rotating Bodies

• Which will have the greater initial velocity?
  Scenario A or B?

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