Advanced Rotational Dynamics for AP Physics

1
Advanced Rotational Dynamicsfor AP Physics

• Common Examples of Rotation
• Review of Introductory Rotational Dynamics
• Combined translation and rotation
• Rolling without slipping
• Rolling with slipping
• Rotational form of Conservation Laws
• Collisions involving objects having Moment of
  Inertia

2
Advanced Rotational Dynamicsfor AP Physics

• Common Examples of Rotation
• Review of Introductory Rotational Dynamics
• Combined translation and rotation
• Rolling without slipping
• Rolling with slipping
• Rotational form of Conservation Laws
• Collisions involving objects having Moment of
  Inertia

3
Advanced Rotational Dynamicsfor AP Physics

• Common Examples of Rotation
• Review of Introductory Rotational Dynamics
• Combined translation and rotation
• Rolling without slipping
• Rolling with slipping
• Rotational form of Conservation Laws
• Collisions involving objects having Moment of
  Inertia

4
Advanced Rotational Dynamicsfor AP Physics

• Common Examples of Rotation
• Review of Introductory Rotational Dynamics
• Combined translation and rotation
• Rolling without slipping
• Rolling with slipping
• Rotational form of Conservation Laws
• Collisions involving objects having Moment of
  Inertia

5
Examples of rotation Earth

• To find the direction of omega, apply the right
  hand rule as follows
• With fingers curling in direction of rotation,
  the thumb gives direction of omega
• i.e. direction of earths omega is upward

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rotation of the sky

• star trails
• centered on Polaris
• rotate once a day

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rotation can be fun
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sometimes the goal is rotational equilibrium

• 1st condition for equilibrium
• Fnet 0
• 2nd condition for equilbrium
• torque net 0
• i.e., torque ccw torque cw

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rotational equilibrium again
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sometimes the goal is large rotational velocity
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M31 rotates once every few hundred million years
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Katrina rotated
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a generic kidney shaped object rotates about a
fixed axisa thing of beauty is a joy forever!
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a turbine (driven by moving fluid) rotates about
a fixed axis
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a jet engine rotates about a fixed axis
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(No Transcript)
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steam driven power plant turbineimagine this
thing rotating at 60 hzgenerating your
electricity
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electric motors are backwards connected
generatorsthey are still mechanical rotators
about a fixed axis
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electric motors power our elderly friends
scooters
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and nuclear subs also(when they are submerged)
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gears rotate about a fixed axis
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gear attached to an electric motor(sounds like a
good idea to me)
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nanotechology electric motor gear (the gear
teeth are smaller than a red blood cell)rotating
about a fixed axis, imaged with an electron
microscope(end of examples of rotation)
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Advanced Rotational Dynamicsfor AP Physics

• Examples of Rotation
• Review of Introductory Rotational Dynamics
• Combined translation and rotation
• Rolling without slipping
• Rolling with slipping
• Rotational form of Conservation Laws
• Collisions involving objects having Moment of
  Inertia

25
Advanced Rotational Dynamicsfor AP Physics

• Examples of Rotation
• Review of Introductory Rotational Dynamics
• Combined translation and rotation
• Rolling without slipping
• Rolling with slipping
• Rotational form of Conservation Laws
• Collisions involving objects having Moment of
  Inertia

26
Advanced Rotational Dynamicsfor AP Physics

• Examples of Rotation
• Review of Introductory Rotational Dynamics
• Combined translation and rotation
• Rolling without slipping
• Rolling with slipping
• Rotational form of Conservation Laws
• Collisions involving objects having Moment of
  Inertia

27
Rotational Kinematics

• T angular position wrt arbitrary origin
• ?? angular displacement (rad)
• ? ?? / ?t d? / dt (rad/s)
• a ? ? / ?t d ? / dt d2? / dt2
  (rad/s2)
• s r ?
• v r ?
• a r a

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a constant implies

• ?? ?0t ½ at2
• ? ?0 a t
• ?2 ?02 2 a ??
• ?? ½ (?0 ?) t
• If a is variable, you need calculus

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Intro Rotational Dynamics

• ? t r x F
• I S mi ri2 (collection of point masses)
• ? r2 dm (continuous matter distribution)
• I total I 1 I 2 I 3 (composite
  object)
• Fnet ma becomes ?net tnet I a

30
Advanced Rotational Dynamicsfor AP Physics

• Common Examples of Rotation
• Review of Introductory Rotational Dynamics
• Combined translation and rotation
• Rolling without slipping
• Rolling with slipping
• Rotational form of Conservation Laws
• Collisions involving objects having Moment of
  Inertia

31
Advanced Rotational Dynamicsfor AP Physics

• Common Examples of Rotation
• Review of Introductory Rotational Dynamics
• Combined translation and rotation
• Rolling without slipping
• Rolling with slipping
• Rotational form of Conservation Laws
• Collisions involving objects having Moment of
  Inertia

32
Advanced Rotational Dynamicsfor AP Physics

• Common Examples of Rotation
• Review of Introductory Rotational Dynamics
• Combined translation and rotation
• Rolling without slipping
• Rolling with slipping
• Rotational form of Conservation Laws
• Collisions involving objects having Moment of
  Inertia

33
Earth revolves (translation of cm) and rotates
about cm
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Look Ma, no hands
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combined translation and rotation

• Ktotal Ktranslational Krotational
• Kof the cm Karound the cm
• ½ mv2 ½ I ?2

36
Advanced Rotational Dynamicsfor AP Physics

• Common Examples of Rotation
• Review of Introductory Rotational Dynamics
• Combined translation and rotation
• Rolling without slipping
• Rolling with slipping
• Rotational form of Conservation Laws
• Collisions involving objects having Moment of
  Inertia

37
Advanced Rotational Dynamicsfor AP Physics

• Common Examples of Rotation
• Review of Introductory Rotational Dynamics
• Combined translation and rotation
• Rolling without slipping
• Rolling with slipping
• Rotational form of Conservation Laws
• Collisions involving objects having Moment of
  Inertia

38
Advanced Rotational Dynamicsfor AP Physics

• Common Examples of Rotation
• Review of Introductory Rotational Dynamics
• Combined translation and rotation
• Rolling without slipping
• Rolling with slipping
• Rotational form of Conservation Laws
• Collisions involving objects having Moment of
  Inertia

39
linear and angular velocity and acceleration are
proportional
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rolling without slipping in our golden years
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rolling without slipping

• v ? r (use with energy conservation)
• atangential a r (use with 2nd laws)
• friction acts, but does no work
• energy conserved as Wnc 0

42
Advanced Rotational Dynamicsfor AP Physics

• Common Examples of Rotation
• Review of Introductory Rotational Dynamics
• Combined translation and rotation
• Rolling without slipping (pure rolling)
• Rolling with slipping
• Rotational form of Conservation Laws
• Collisions involving objects having Moment of
  Inertia

43
Advanced Rotational Dynamicsfor AP Physics

• Common Examples of Rotation
• Review of Introductory Rotational Dynamics
• Combined translation and rotation
• Rolling without slipping (pure rolling)
• Rolling with slipping
• Rotational form of Conservation Laws
• Collisions involving objects having Moment of
  Inertia

44
Advanced Rotational Dynamicsfor AP Physics

• Common Examples of Rotation
• Review of Introductory Rotational Dynamics
• Combined translation and rotation
• Rolling without slipping (pure rolling)
• Rolling with slipping
• Rotational form of Conservation Laws
• Collisions involving objects having Moment of
  Inertia

45
linear and angular velocities and accelerations
are independent,i.e., hes not getting very much
bang (v) for his buck (?)
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dont try this at home
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rolling with slipping

• v ? ? r
• atangential ? a r
• apply Fnet ma to find atangential of the cm
• apply ?net Ia to find a around the cm
• to compute t where pure rolling sets in,
• set a(t) a(t) r, where a(t) and a(t) are
• solutions of force and torque equations

48
Advanced Rotational Dynamicsfor AP Physics

• Common Examples of Rotation
• Review of Introductory Rotational Dynamics
• Combined translation and rotation
• Rolling without slipping
• Rolling with slipping
• Rotational form of Conservation Laws
• Collisions involving objects having Moment of
  Inertia

49
Advanced Rotational Dynamicsfor AP Physics

• Common Examples of Rotation
• Review of Introductory Rotational Dynamics
• Combined translation and rotation
• Rolling without slipping
• Rolling with slipping
• Rotational form of Conservation Laws
• Collisions involving objects having Moment of
  Inertia

50
Advanced Rotational Dynamicsfor AP Physics

• Common Examples of Rotation
• Review of Introductory Rotational Dynamics
• Combined translation and rotation
• Rolling without slipping
• Rolling with slipping
• Rotational form of Conservation Laws
• Collisions involving objects having Moment of
  Inertia

51
definition of angular momentum

• L r x p r x mv (point m moving at v)
• L I ? (rigid body with moment I)

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law of motion in terms of L

• torque net ?L/ ?t dL / dt

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Conservation of angular momentum

• if torque net 0, then dL / dt 0
• and so, L constant
• so, I initial ? initial I final ? final
  
• (ice skater)

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easier said than done
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The agony of defeat(net torque ? 0, L not
conserved)
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The thrill of victory(net torque 0, L
conserved)
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Conservation of energy, including rotation

• Simply allow for rotational
• as well as translational K

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Conservation of linear momentum

• is not affected by rotational motion, but
  applies to the cm of the body as before
• (i.e., if Fnet,ext 0 on system. That is,
• if something is nailed down and rotates,
  linear momentum is not conserved, but L can be)

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Advanced Rotational Dynamicsfor AP Physics

• Common Examples of Rotation
• Review of Introductory Rotational Dynamics
• Combined translation and rotation
• Rolling without slipping
• Rolling with slipping
• Rotational form of Conservation Laws
• Collisions involving objects having Moment of
  Inertia

60
Advanced Rotational Dynamicsfor AP Physics

• Common Examples of Rotation
• Review of Introductory Rotational Dynamics
• Combined translation and rotation
• Rolling without slipping
• Rolling with slipping
• Rotational form of Conservation Laws
• Collisions involving objects having Moment of
  Inertia

61
Advanced Rotational Dynamicsfor AP Physics

• Common Examples of Rotation
• Review of Introductory Rotational Dynamics
• Combined translation and rotation
• Rolling without slipping
• Rolling with slipping
• Rotational form of Conservation Laws
• Collisions involving objects having Moment of
  Inertia

62
Advanced Rotational Dynamicsfor AP Physics

• Common Examples of Rotation
• Review of Introductory Rotational Dynamics
• Combined translation and rotation
• Rolling without slipping
• Rolling with slipping
• Rotational form of Conservation Laws
• Collisions (and other systems) involving objects
  having Moment of Inertia

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• this is rather important on the AP Exam

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• Unfortunately, few common practical applications
  exist for this type of problem.
• They are simply artificially contrived

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Any or all 3 conservation laws of energy, linear
momentum, and angular momentum may be
satisfied!You have to figure out which ones are
operating, if they dont tell you!

• If Wnc 0, then Einitial Efinal
• If Fnet,ext 0, then pinitial pfinal
• If ?net 0, then Linitial Lfinal

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The End

• Thats it! Now go make a 5!