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Week 6: Vector Torque and Angular Momentum |
•For rigid objects (or collections of point particles) that have mirror symmetry across the axis of rotation and/or mirror symmetry across the plane of rotation, the vector angular momentum can be written in terms of the scalar moment of inertia about the axis of rotation (defined and used in week 5) and the vector angular velocity as:
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L = Iω~
• For rigid objects or collections of point particles that lack this symmetry with respect to an axis of rotation (direction of ω~)
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L =6 Iω~
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for any scalar I. In general, L precesses around the axis of rotation in these cases and requires a constantly varying nonzero torque to drive the precession.
•When two (or more) isolated objects collide, both momentum and angular momentum is conserved. Angular momentum conservation becomes an additional equation (set) that can be used in analyzing the collision.
•If one of the objects is pivoted, then angular momentum about this pivot is conserved but in general momentum is not conserved as the pivot itself will convey a significant impulse to the system during the collision.
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• Radial forces – any force that can be written as F = Fr~r – exert no torque on the masses that they act on. Those object generally move in not-necessarily-circular orbits with constant angular momentum.
• When a rapidly spinning symmetric rotator is acted on by a torque of constant magnitude that is (always) perpendicular to the plane formed by the angular momentum and a vector in second direction, the angular momentum vector precesses around the second vector. In
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particular, for a spinning top with angular momentum L tipped at an angle θ to the vertical, the magnitude of the torque exerted by gravity and the normal force on the top is:
τ = |D~ × mgzˆ| = mgD sin(θ) = |
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or
ωp = mgD L
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In this expression, ωp is the angular precession frequency of the top and D is the vector from the point where the tip of the top rests on the ground to the center of mass of the top. The direction of precession is determined by the right hand rule.
6.1: Vector Torque
In the previous chapter/week we saw that we could describe rigid bodies rotating about a single axis quite accurately by means of a modified version of Newton’s Second Law:
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(565) |
τ = rF F sin(φ) = |~rF × F | = Iα |
where I is the moment of inertia of the rigid body, evaluated by summing/integrating:
I = |
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miri2 = Z |
r2dm |
(566) |
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In the torque expression ~rF is a was a vector in the plane perpendicular to the axis of rotation leading from the axis of rotation to the point where the force was applied. r in the moment of inertia