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Physics 116 Lectures -- part 3

# Physics 116 Lectures -- part 3 - Feb 20 2010 Last lecture...

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Feb. 20, 2010 Last lecture: Conservation of energy Today: Center of mass Newton’s 2 nd law for a system of particles Next lecture: Momentum Collision and impulse Hour exam 2: Monday March 1, 2010, cover chapter 5-8.

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An elevator is rising at constant speed. Which of the following statement is not true: a) The upward cable force is constant b) The kinetic energy of the elevator is constant c) The gravitational potential energy of the Earth- elevator system is constant d) The acceleration of the elevator is zero e) The total energy of the Earth-elevator system is constant.
For complex bodies, there is one point, the Center of Mass , which moves under the influence of external forces as if it were a particle with all mass concentrated at that point. Center of Mass The center of mass for an object is always located on a line of symmetry at the physical ‘balance point’ It can change when the configuration of the body change.

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Finding the Center of Mass N i i i sys com r m M r 1 1 For a system of discrete bodies eg. Two body system in 2-dimension 2 1 2 2 1 1 1 2 1 2 2 1 1 1 1 1 m m y m y m y m M y m m x m x m x m M x N i i i sys com N i i i sys com
Finding the Center of Mass cm m m m x m x m x m x com 83 4 . 3 5 . 2 2 . 1 70 4 . 3 140 5 . 2 0 2 . 1 3 2 1 3 3 2 2 1 1 m 1 =1.2kg, m 2 =2.5kg, m 3 =3.4kg, Where is the center of mass? cm m m m y m y m y m y com 58 4 . 3 5 . 2 2 . 1 120 4 . 3 0 5 . 2 0 2 . 1 3 2 1 3 3 2 2 1 1

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1 dm r M r sys com For continuous bodies , 1 , 1 , 1 or , 1 zdV V z ydV V y xdV V x dV r V r com com com com With a constant mass density =M/V Finding the Center of Mass
Finding the Center of Mass A uniform metal plate P of radius 2R from which a disk of radius R is stamped out. Where is the center of mass? R R R R R m m x x m m x m x m x P S S P S P S S P P S P 3 1 ) 2 ( ) ( 0 2 2 2  Key idea: In finding COM, an object can be replaced by a point with all its mass concentrate at its COM. Key idea: From symmetry, the COM must be on the x axis, or y com =0. The COM of the plate+disk is located at the center.

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Newton’s 2 nd Law For a system of particles, F net is the net force of all external forces. M is the total mass of the system. a com is the COM acceleration. com net a M F net n n com n n com n n com F F F F a m a m a m a M V m V m V m V M r m r m r m r M 2 2 1 2 2 1 1 2 2 1 1 2 2 1 1 Proof:
The center of mass of a system of particles has constant velocity if: A) the forces exerted by the particles on each other sum to zero B) the external forces acting on particles of the system sum to zero

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Physics 116 Lectures -- part 3 - Feb 20 2010 Last lecture...

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