Chapt. 10 Angular Momentum

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Chapt. 10 Angular Momentum Definition of angular momentum Vector nature of torque 06/09/22 1 Phys 201, Spring 2011
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Chapt. 10 Angular Momentum. Definition of angular momentum Vector nature of torque. F t. Angular rotation using vectors. Angular quantities in vector notation. τ = F r sin ϕ = F L. The vector product. It follows: Non commutative: And distribution rule: Unit vectors:. - PowerPoint PPT Presentation

Transcript of Chapt. 10 Angular Momentum

  • Chapt. 10Angular MomentumDefinition of angular momentumVector nature of torque**Phys 201, Spring 2011

    Phys 201, Spring 2011

  • Angular rotation using vectorsAngular quantities in vector notation = F r sin = F L**Phys 201, Spring 2011

    Phys 201, Spring 2011

  • The vector product

    It follows: Non commutative:

    And distribution rule:

    Unit vectors:

    **Phys 201, Spring 2011

    Phys 201, Spring 2011

  • Example: Vector algebraIf and = 12 find .

    Let

    Then

    Finally:**Phys 201, Spring 2011

    Phys 201, Spring 2011

  • Linear momentum --> Angular momentumThe linear momentum

    The angular momentum with respect to the axis of rotation must scale with r:

    m= r m v = m r2 = I **Phys 201, Spring 2011

    Phys 201, Spring 2011

  • Linear momentum --> Angular momentumThe angular momentum, the vector nature: **Phys 201, Spring 2011

    Phys 201, Spring 2011

  • i j Angular momentum of a rigid bodyabout a fixed axis:Consider a rigid distribution of point particles rotating in the x-y plane around the z axis, as shown below. The total angular momentum around the origin is the sum of the angular momenta of each particle:r1r3r2 m2 m1m3 v2 v1v3We see that L is in the z direction.Using vi = ri , we get

    (since ri and vi are perpendicular) **Phys 201, Spring 2011

    Phys 201, Spring 2011

  • Angular momentum of a rigid bodyabout a fixed axis:In general, for an object rotating about a fixed (z) axis we can write LZ = I The direction of LZ is given by the right hand rule (same as ). We will omit the axis (Z) subscript for simplicity, and write L = I z**Phys 201, Spring 2011

    Phys 201, Spring 2011

  • Rotational and LinearQuantityLinearAngular PositionVelocityAccelerationTimeInertiaDynamicsMomentumKinetic energy

    **Phys 201, Spring 2011

    Phys 201, Spring 2011

  • The rotational analogue of force F is torque Define the rotational analogue of momentum p to be angular momentumThe 2nd Law in rotation:Translational (linear) motion for a system of particles

    What is the rotational version of this?? F = m a**Phys 201, Spring 2011

    Phys 201, Spring 2011

  • Definitions & Derivations...First consider the rate of change of L:So:Recall:EXT Finally:Direct analogue:**Phys 201, Spring 2011

    Phys 201, Spring 2011

  • What does it mean? where andIn the absence of external torquesTotal angular momentum is conserved**Phys 201, Spring 2011

    Phys 201, Spring 2011

  • L (and therefore the wheel)moves in a horizontalcircle around O: precessionGyroscope:

  • The precession frequency

    Lets calculate the precession frequency L forms a circular motion:(Last figure)

  • Question: We repeat the gyroscope experiment on the moon (g_moon = 1/6 g_Earth) but with an angular velocity double from the one in the lecture.Will the precession of the wheel be faster, slower or the same?

    ******