Rolling, Torque, and Angular Momentum

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CHAPTER-11

• Rolling, Torque, and Angular Momentum

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Ch 11-2 Rolling as Translational and Rotation
Combined

• Rolling Motion
• Rotation of a rigid body about an axis not fixed
  in space
• Smooth Rolling
• Rolling motion without slipping
• Motion of com O and point P
• When the wheel rotates through angle ?, P moves
  through an arc length s given by
• sR ?
• Differentiating with respect to t
• We get ds/dt R d?/dt
• vcom R?

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Ch 11-2 Rolling as Translational and Rotation
Combined

• Rolling motion of a rigid body
• Purely rotational motion Purely
  translational mption
• Pure rotational motion all points move with same
  angular velocity ?.
• Points on the edge have velocity vcom R?
• with vtop vcom and vbot -
  vcom
• Pure translational motion All points on the
  wheel move towards right with same velocity vcom

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Ch-11 Check Point 1

• The rear wheel on a clowns bicycle has twice the
  radius of the front wheel.
• (a) When the bicycle is moving , is the linear
  speed at the very top of the rear wheel greater
  than, less than, or the same as that of the very
  top of the front wheel?
• (b) Is the angular speed of the rear wheel
  greater than, less than, or the same as that of
  the front wheel?

• 1. (a) vtop-frontvtop-rear2 vcom
• same
• (b) vtop-front vtop-rear
• 2?frontRfront 2?rearRrear
• ?rear/ ?front Rfront /Rrear
• Rrear 2 Rfront
• ?rear/ ?front Rfront /Rrear 1/2
• ?rear lt ?front
• less

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Ch 11-3 Kinetic Energy of Rolling

• Rolling as a Pure Rotation about an axis through
  P
• Kinetic energy of rolling wheel rotating about an
  axis through P
• K (IP ?2)/2
• where IP IcomMR2 and R? vcom
• K (IP ?2)/2 (Icom ?2 MR2 ?2)/2
• K (Icom ?2)/2 (Mv2com)/2
• K KRotKTrans

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Ch 11-4 The Forces of Rolling

• In smooth rolling, static frictional force fs
  opposes the sliding force at point P
• VcomR?
• d/dt(Vcom)d/dt(R?)
• acomR d?/dtR?
• Accelerating Torque acting clockwise static
  frictional force fs tendency to rotate counter
  clockwise

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Ch 11-4-cont. Rolling Down a Ramp

• Rigid cylinder rolling down an incline plane,
  acom-x?
• Components of force along the incline plane
  (upward) and perpendicular to plane
• Sliding force downward-static friction force
  upward opposite trends
• fs-Mgsin?Macom-x acom-x (fs/M)-gsin?
• To calculate fs apply Newtons Second Law for
  angular motion Net torque I?
• Torque of fs about body com fsR I?
• But ?-acom-x/R then
• fs Icom?/R-Icomacom-x/R2
• acom-x(fs/M)-gsin?
• (-Icomacom-x/MR2)- gsin?
• acom-x (1Icom /MR2) - gsin?
• acom-x - gsin?/(1Icom /MR2)

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Ch-11 Check Point 2

• Disk A and B are identical and rolls across a
  floor with equal speeds. The disk A rolls up an
  incline, reaching a maximum height h, and disk B
  moves up an incline that is identical except that
  is frictionless. Is the maximum height reached by
  disk B greater than, less than or equal to h?

• A is rolling and its kinetic energy before decent
  
• KA Icom?2 /2 M(vcom)2/2
• KB M(vcom)2/2
• vBltvA
• Height h of incline, given by conservation of
  mechanical energy
• ?K - ?Ug hv2/2g
• hBlthA because vBltvA

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Ch 11-5 The Yo-Yo

• Yo-Yo is Physics teaching Lab.
• Yo-Yo rolls down its string for a distance h
  and then climbs back up.
• During rolling down yo-yo loses potential energy
  (mgh) and gains translational kinetic energy
  (mv2com/2) and rotational kinetic energy (
  Icom?2/2).
• As it climbs up it loses translational kinetic
  energy and gains potential energy .
• For yo-yo, equations of incline plane modify to
  ?90
• acom- g/(1Icom /MR02)

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Ch 11-6 The Torque Revisited

• ?r xF
• ?r Fsin?
• ? r F? r? F
• Vector product
• ?r xF
• ??i j k ?
• ?x y z ?
• ?Fx Fy Fz?

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Ch-11 Check Point 3

• The position vector r of a particle points along
  the positive direction of a z-axis. If the torque
  on the particle is (a) zero
• (b) in the negative direction of x and
• (c) in the negative direction of y, in what
  direction is the force producing the torque

• ?rxFrfsin?
• ?rfsin? 0 (?0, 180)
• i k x F, i.e. F along j
• (c) jk x F i.e. F along -i

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Ch-11 Check Point 4

• In part a of the figure, particles 1 and 2 move
  around point O in opposite directions, in circles
  with radii 2m and 4m . In part b, particles 3 and
  4 travel in the same direction along straight
  lines at perpendicular distance of 4m and 2m from
  O. Particle 5 move directly away from O.
• All five particles have the same mass and same
  constant speed.
• (a) Rank the particles according to magnitude of
  their angular ,momentum about point O, greatest
  first
• (b) which particles have negative angular
  momentum about point O.

l r?mv r? 4m for 1 and 3 2m for 2 and 4
0 for 5 Ans (a) 1 and 3 tie, then 2 and 4
tie, then 5 (zero) (b) 2 and 3
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Ch 11-7,8,9 Angular Momentum

• l r x p rp sin?
• r p? r? p
• Newtons Second Law
• Fnet dp/dt ?net dl/dt
• For system of particles
• L?li ?net dL/dt

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Ch-11 Check Point 5

• The figure shows the position vector r of a
  particle at a certain instant, and four choices
  for the directions of force that is to accelerate
  the particle. All four choice lie in the xy
  plane.
• (a) Rank the choices according to the magnitude
  of the time rate of change (dl/dt) they produce
  in the angular momentum f the particle about
  point O, greatest first
• (b) Which choice results in a negative rate of
  change about O?

• ? (dl/dt)rxF
• ?1 ?3 rxF1 rxF3
• and ?2 ?4 0

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Ch 11-7 Angular Momentum of a Rigid Body Rotating
about a Fixed Axis

• Magnitude of angular momentum of mass ?mi
• li ri x pi ri pi sin90 ri ?mivi
• li ? ( ri and pi)
• Component of li along Z-axis
• liZ li sin ? ri sin90 ?mivir?i ?mivi
• vi r?i ?
• liZr?i ?mivir?i ?mi (r?i ?)r?i 2?mi ?
• Lz ? liZ (?r?i 2?mi ) ?I ?
• (rigid body fixed axis)

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Ch-11 Check Point 6

• In the figure, a disk, a hoop and a solid sphere
  are made to spin about fixed central axis (like a
  top) by means of strings wrapped around them,
  with the string producing the same constant
  tangential force F on all three objects. The
  three objects have the same mass and radius, and
  they are initially stationary. Rank the objects
  according to
• (a) angular momentum about their central axis
• (b) their angular speed, greatest first, when the
  string has been pulled for a certain time t.

• ?net dl/dtFR l ?net x t
• Since ?net FR for all three objects,
  lhoopldisklsphere
• ?f?i?t ?netI?FR ?FR/I
• ?i0 ?f?i?t ?tFRt/I
• ?f?tFRt/I
• IhoopMR2 IDiskMR2/2
• Isphere 2/5 MR2
• ?f-hoop FRt/Ihoop FRt/MR2
• ?f-Disk FRt/IDisk 2(FRt/MR2)
• ?f-Sphere FRt/ISphere 5(FRt/MR2)/2
• Sphere, Disk and hoop angular speed

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Ch 11-11 Conservation of Angular momentum

• Newtons Second Law in angular form
• ?net dL/dt
• If ?net 0 then
• L a constant (isolated system)
• Law of conservation of angular momentum
• Li L
• Ii ?i If ?f