PHY183-Lecture29 - Physics for Scientists Engineers 1...

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March 6, 2006 Physics for Scientists&Engineers 1 1 Physics for Scientists & Physics for Scientists & Engineers 1 Engineers 1 Spring Semester 2006 Lecture 29
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March 6, 2006 Physics for Scientists&Engineers 1 2 Position of the Center of gravity Position of the Center of gravity Three scalar equations for the three coordinates: Particular case: homogeneous objects = constant density Steps to find the center of mass: Reduce the amount of work: find symmetry planes => the C.M. lies within the symmetry planes Pick the right set of coordinates to describe your object (cartesian, cylindrical, spherical…) According to this choice: use the right expression for dV Work out the boundaries of the triple integral shape of the object 1 1 1 ( ) ; ( ) ; ( ) V V V X x r dV Y y r dV Z z r dV M M M ! ! ! = = = " " " r r r
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March 6, 2006 Physics for Scientists&Engineers 1 3 Motion of Extended Objects Motion of Extended Objects The motion of extended objects can seem complicated But if we follow the center of gravity instead of the whole wrench Things are simpler: motion of center of gravity + rotation around it
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March 6, 2006 Physics for Scientists&Engineers 1 4 Center of Mass Motion and Momentum Center of Mass Motion and Momentum The center of mass is important, even in cases where there is no gravitational force. (no “center of gravity”) Still relevant to inertial mass We can find the velocity of the center of mass starting from the position of the center of mass Take the derivative with respect to time to get the velocity Total Momentum is proportional to center-of-mass velocity: 1 1 n i i i R rm M = = ! r r 1 1 1 1 1 1 1 1 n n n n i i i i i i i i i i i d d d V R rm m r m v p dt dt M M dt M M = = = = ! " # = = = = $ % & ( ( ( ( r r r r r r 1 n i i P MV p = = = ! r r r
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March 6, 2006 Physics for Scientists&Engineers 1 5 Newton Newton’ s Second law for the Center of Mass s Second law for the Center of Mass We can take the time derivative of the momentum to get the force Note that forces exerted by particles on other particles of the same system must sum to zero, so we have where is the acceleration of the center of mass 1 1 1 ( ) n n n i i i i i i d d d d P MV p p F dt dt dt dt = = = ! " = = = = # $ % & r r r r r d dt r P = M r A = r F net r A
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March 6, 2006 Physics for Scientists&Engineers 1 6 Quiz 29.1 Quiz 29.1 Two astronauts (m 1 =80 kg, m 2 =120 kg) are tethered together in outer space and are at rest with respect to the earth. If astronaut 1 starts to pull
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