Angular Momentum 8 01 W10D1 Today s Reading Assignment MIT 8 01 Course Notes Chapter 19 Angular Momentum Sections 19 1 19 6 Announcements Problem Set 7 due Week 10 Tuesday at 9 pm in box outside 26 152 Math Review Week 10 Tuesday at 9 pm in 26 152 Exam 3 Tuesday Nov 26 7 30 9 30 pm Conflict Exam 3 Wednesday Nov 27 8 10 am 10 12 noon Nov 27 Drop Date Overview Torque and Angular Momentum Angular momentum is defined as L S rS p Torque about a point S is equal to the time derivative of the angular momentum about S dL S S dt When torque about S is zero angular momentum is constant L S f L S i Angular Momentum of a Point Particle Point particle of mass m moving with a velocity v p mv Momentum Fix a point S Vector rS from the point S to the location of the object Angular momentum about the point S SI Unit kg m 2 s 1 L S rS p Cross Product Angular Momentum of a Point Particle Magnitude L S rS p L S rS p sin a moment arm rS rS sin L S rS p b perpendicular momentum pS p sin L S rS p Angular Momentum of a Point Particle Direction Direction Right Hand Rule Concept Q Mag of Angular Momentum In the above situation where a particle is moving in the x y plane with a constant velocity the magnitude of the angular momentum about the point S the origin 1 decreases then increases 2 increases then decreases 3 is constant 4 is zero because this is not circular motion Table Problem Angular Momentum and Cross Product A particle of mass m moves with a uniform velocity v vx i vy j At time t the position vector of the particle with respect to the point S is rS x i y j Find the direction and the magnitude of the angular momentum about the origin the point S at time t Angular Momentum and Circular Motion of a Point Particle Fixed axis of rotation z axis Angular velocity z k Velocity v v R z Angular momentum about the point S L S rS p rS mv Rmv k mR 2 z k Table Prob Angular momentum Along Axis of Rotation for Circular Motion A particle of mass m moves in a circle of radius R at an angular speed about the z axis in a plane parallel to but a distance h above the x y plane a Find the magnitude and the direction of the angular momentum L 0 relative to the origin b Find the z component of L 0 Hint Use r0 rr hk Concept Q Direction of Ang Mom A particle of mass m moves in a circle of radius R at an angular speed about the z axis in a plane parallel to but above the x y plane Relative to the origin 1 L 0 is constant 2 L 0 is constant but direction of L 0 is not 3 Direction of L 0 is constant but L 0 is not 4 L 0 has no z component Worked Example Angular Momentum of Two Particles Two identical particles of mass m move in a circle of radius r 180 out of phase at an angular speed about the z axis in a plane parallel to but a distance h above the x y plane a Find the magnitude and the direction of the angular momentum L 0 relative to the origin b Is this angular momentum relative to the origin constant If yes why If no why is it not constant Worked Example Angular Momentum of Two Particles r0 1 xi yj hk rr 1 hk r x 2 y 2 1 2 v v r L 0 1 r0 1 mv rr 1 hk mr L 0 1 mr 2 k hmr r 1 L 0 2 r0 2 mv rr 2 hk mr mr 2 k hmr r 2 r 1 r 2 L 0 1 L 0 2 mr 2 k hmr r 1 mr 2 k hmr r 2 2mr 2 k Angular Momentum of a Ring A circular ring of radius R and mass M rotates at an angular speed about the zaxis in a plane parallel to but a distance h above the x y plane Find the magnitude and the direction of the angular momentum L 0 relative to the origin Divide ring into pairs of small objects with mass L 0 1 L 0 2 2 mr 2 k L0 2 2 m r k Mr k pair pairs Concept Q Symmetric Body A rigid body with rotational symmetry body rotates at a constant angular speed about it symmetry z axis In this case 1 L 0 is constant 2 L 0 is constant but L L is not 0 0 3 L L is constant but L is not 0 0 0 4 L 0 has no z component 5 Two of the above are true Angular Momentum of Cylindrically Symmetric Body A cylindrically symmetric rigid body rotating about its symmetry axis at a constant angular velocity z k with z 0 has angular momentum about any point on its axis Lz dLz body dm r z k body 2 dm rdm z k body I z z k 2 dm Kinetic Energy of Cylindrically Symmetric Body A cylindrically symmetric rigid body with moment of inertia Iz rotating about its symmetry axis at a constant angular velocity z k z 0 Angular Momentum Kinetic energy K rot Lz I z z 2 L 1 I z z2 z 2 2I z Concept Q Angular Momentum of Disk A disk with mass M and radius R is spinning with angular speed about an axis that passes through the rim of the disk perpendicular to its plane The magnitude of its angular momentum is 1 1 M R 2 2 4 2 1 M R 2 2 2 3 3 M R 2 2 2 1 4 M R 2 4 5 1 M R 2 2 3 6 M R 2 2 Concept Q Non Symmetric Body A body rotates with constant angular speed about the z axis which is not a symmetry axis of the body Relative to the origin 1 L 0 2 L 0 is constant is constant but L 0 L 0 is not 3 L 0 L 0 is constant but L 0 is not 4 L 0 has no z component Table Problem Angular Momentum of a Two Particles About Different Points Two point like particles 1 and 2 each of mass m are rotating at a constant angular speed about point A How does the angular momentum about the point B compare to the angular momentum about point A What about at a later time when the particles have rotated by 90 degrees Conservation of Angular Momentum Time Derivative of Angular Momentum for a Point Particle Angular momentum of a point particle about S L S rS p d Time derivative of the …
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