Chapt. 10 Angular Momentum Definition of angular momentum Vector nature of torque 11/17/2018 Phys 201, Spring 2011

Angular rotation using vectors Angular quantities in vector notation Ft τ = F r sin ϕ = F L 11/17/2018 Phys 201, Spring 2011

The vector product It follows: Non commutative: And distribution rule: Unit vectors: 11/17/2018 Phys 201, Spring 2011

Example: Vector algebra If and = 12 find . Let Then Finally: 11/17/2018 Phys 201, Spring 2011

Linear momentum --> Angular momentum The linear momentum The angular momentum with respect to the axis of rotation must scale with r: m = r m v = m r2 ω = I ω 11/17/2018 Phys 201, Spring 2011

Linear momentum --> Angular momentum The angular momentum, the vector nature: 11/17/2018 Phys 201, Spring 2011

Angular momentum of a rigid body about 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 ri and vi are perpendicular) v1 We see that L is in the z direction. m2 j Using vi = ω ri , we get r2 r1 m1 i v2 ϕ r3 m3 v3 11/17/2018 Phys 201, Spring 2011

Angular momentum of a rigid body about 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 the right hand rule (same as ω). We will omit the axis (Z) subscript for simplicity, and write L = I ω z ω 11/17/2018 Phys 201, Spring 2011

Rotational and Linear Quantity Linear Angular Position Velocity Acceleration Time Inertia Dynamics Momentum Kinetic energy 11/17/2018 Phys 201, Spring 2011

The 2nd Law in rotation: F = m a 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 11/17/2018 Phys 201, Spring 2011

Definitions & Derivations... First consider the rate of change of L: So: Recall:  τEXT Finally: Direct analogue: 11/17/2018 Phys 201, Spring 2011

What does it mean? where and In the absence of external torques Total angular momentum is conserved 11/17/2018 Phys 201, Spring 2011

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

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

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?