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Chapt. 10 Angular Momentum

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:.

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Chapt. 10 Angular Momentum

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  1. Chapt. 10Angular Momentum Definition of angular momentum Vector nature of torque Phys 201, Spring 2011

  2. Ft Angular rotation using vectors • Angular quantities in vector notation • τ = F r sin ϕ • = F L Phys 201, Spring 2011

  3. The vector product • It follows: Non commutative: • And distribution rule: Unit vectors: Phys 201, Spring 2011

  4. Example: Vector algebra If and = 12 find . Let Then Finally: Phys 201, Spring 2011

  5. Linear momentum --> Angular momentum m • The linear momentum • The angular momentum with respect to the axis of rotation must scale with r: = r m v = m r2 ω = I ω Phys 201, Spring 2011

  6. Linear momentum --> Angular momentum • The angular momentum, the vector nature: Phys 201, Spring 2011

  7. 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: (since riand vi are perpendicular) v1 We see that L is in the z direction. m2 j Using vi = ω ri, we get r2 m1 r1 i v2 ϕ r3 m3 v3 Phys 201, Spring 2011

  8. 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 ω • The direction of LZ is given by theright hand rule (same as ω). • We will omit the axis (Z) subscript for simplicity, and write L= I ω z ω Phys 201, Spring 2011

  9. Rotational and Linear Quantity Linear Angular Position Velocity Acceleration Time Inertia Dynamics Momentum Kinetic energy Phys 201, Spring 2011

  10. The 2nd Law in rotation: • Translational (linear) motion for a system of particles • What is the rotational version of this?? F = m a • The rotational analogue of force F is torque  • Define the rotational analogue of momentum p to be angular momentum Phys 201, Spring 2011

  11. Definitions & Derivations... • First consider the rate of change of L: So: Recall:  τEXT Finally: Direct analogue: Phys 201, Spring 2011

  12. What does it mean? • where and • In the absence of external torques Total angular momentum is conserved Phys 201, Spring 2011

  13. Gyroscope: L (and therefore the wheel) moves in a horizontal circle around O: “precession”

  14. Let’s calculate the precession frequency (Last figure) L forms a circular motion: The precession frequency

  15. 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?

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