Angular Momentum

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Angular Momentum

• AP Physics C
• Mrs. Coyle

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Remember Torque

• tr- x F rFsinf

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Cross Product

• The cross product of two vectors AxB is a third
  vector that is perpendicular to the plane of A
  and B (according to the right hand rule).

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Properties of the Cross Product

• 1) AxB-BxA (not commutative)
• 2) AxA0 (or if A is // B, AxB0)
• 3)AxBAB, when A __ B

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Properties of the Cross Product, Contd

• 4) Ax(BC)AxB AxC (distributive)
• 5)

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Properties of Cross Products of Unit Vectors

• ixi0
• jxj0
• kxk0

• ixjk
• jxkI
• kxij

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Using Determinants

• or

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Example 3

• Two vectors are given by A-3i4j and B2i3j.
• Find
• a)AxB
• b) the angle between A and B
• Ans a)-17.0k, b)70.6o

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Angular Momentum, L

• For a particle of mass m at position r and
  linear momentum p.

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Angular Momentum, L r x p

• The instantaneous angular momentum of a
  particle relative to the origin is defined as the
  cross product of the particles instantaneous
  position vector r and its instantaneous linear
  momentum p
• L r x p

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Angular Momentum

• L r mv sin f
• is the angle between r and p .
• Remember pmv

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Rotational Analog of Newtons Second Law

• St and L must be measured about the same origin.
• For origin fixed in an inertial frame.

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Units of Angular Momentum

• SI units (kg.m2)/ s

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Direction of Angular Momentum

• The magnitude and direction of L depend on the
  choice of origin
• The direction of L is perpendicular to the plane
  formed by r and p.

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Example Uniform Circular Motion

• A particle in uniform circular motion has a
  constant angular momentum about an axis through
  the center of its path.

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Angular Momentum of a System of Particles

• The vector sum of the angular momenta of the
  individual particles
• Ltot L1 L2 Ln SLi

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Note

• Any torques associated with the internal forces
  acting in a system of particles are zero.
• So

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Resultant Torque about the CM

• The resultant torque acting on a system about an
  axis through the center of mass equals the time
  rate of change of angular momentum of the system
  regardless of the motion of the center of mass
  (even if the center of mass is accelerating).

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Ex 11

• A light rod 1.00m in length joins two
  particles, with masses 4kg and 3kg at its ends.
  The combination moves in the xy plane about a
  pivot the center of the rod, counterclockwise.
  Determine the angular momentum of the system
  about the origin when the speed of each particle
  is 5m/s.
• Ans 17.5k kg m2 /s