Lect20 - PHYS 172 Modern Mechanics Lecture 20 Angular...

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Fall 2011 PHYS 172: Modern Mechanics Lecture 20 – Angular momentum Read 11.4 – 11.7 1
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The angular momentum principle d ! p dt = ! F net d ! L A dt = ? d ! r A ! ! p ( ) dt = d ! r A dt ! ! p + ! r A ! d ! p dt = ! v !" m ! v 0 = = ! F net The angular momentum principle for a point particle A ! r A ! F net ! p d ! L A dt = ! r A ! ! F net ! ! L A = ! r A " ! F net ( ) ! t torque : ! ! A " ! r A # ! F net = ! A = ! A " t Note: The angular momentum principle is derived from the momentum principle 2
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Torque sin A A net r F ! " # $ $ ! ! ! A A net r F !" # ! ! ! θ =90˚ θ =0˚ BIG TORQUE SMALL TORQUE BIG TORQUE SMALL TORQUE 3
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Example: momentum and angular momentum principles d ! p dt = ! F net ( ) dm v mg dt = dv g dt = Use the momentum principle: d ! L A dt = ! r A ! ! F net = ! " A Use the angular momentum principle: ! ! A A = xmg L A = r ! p = xmv ! L A = ! r A ! ! p ! L A d xmv ( ) dt = xm dv dt dv dt = g Falling object (nonrelativistic) 4
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Conservation of angular momentum ! ! L A , system + ! ! L A , surroundings = 0 Example: Important: both L ` s must be about the same point (axis) 5
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Example: Kepler and elliptical orbits Kepler, 1609: l a radius vector joining any planet to the Sun sweeps out equal areas in equal lengths of time z Can be easily proven using conservation of angular momentum See book p. 430 (11.4) 6
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