dynam1 - Physics 53 Dynamics 1 . from the same principles,...

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Physics 53 Dynamics 1 ... from the same principles, I now demonstrate the frame of the System of the World. — Isaac Newton, Principia Reference frames When we say that a particle moves in a certain way, what we mean is that it is observed to move that way. Our description necessarily concerns what is (or might be) seen by an observer . Different observers may describe the same situation differently. Here we investigate how to make a general framework to reconcile their differences. If the observers are at rest relative to each other, the differences are only those that might arise from choosing different coordinate systems. These are easily taken into account. But if the observers are in motion relative to each other, things are more subtle. Suppose we have two observers, A and B, who are watching the motion of particle P. Each stands at the origin of a set of Cartesian axes used to specify the location of the particle. This set of axes constitutes the reference frame of the observer. At a particular time t , observer A will use the vector r = ( x , y , z ) to represent the particle’s position. At the same time, B will use the vector r = ( x , y , z ) for that purpose. Let the origin of B's frame be located relative to A's frame by the vector R . If the observers are moving relative to each other, R will change with time. The drawing shows the situation. From the vector triangle we see that r = r + R . A careful choice of notation helps. Let r i / j denote the position of object i relative to object j . For the particle P and our two observers we have r = r P / A , r = r P / B , R = r B / A . These are related, as the vector triangle shows, by r P / A = r P / B + r B / A . The ±rst upper subscript on the right ( P ) is the same as the upper subscript on the left, while the second lower subscript on the right ( A ) is the same as the lower subscript on the left. As time elapses these vectors will in general change. The time derivates yield: r r R A B z y x PHY 53 1 Dynamics 1
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Relative velocity formula v P / A = v P / B + v B / A There are many uses for this formula. Taking another time derivative, we have: a P / A = a P / B + a B / A . These three formulas, relating the position, velocity and acceleration of the particle as seen by two observers in motion relative to each other, are called the equations of Galilean relativity , after Galileo, who Frst realized the important role of reference frames. If the observers move with constant velocity relative to each other, so that a B / A = 0 , we have a P / B = a P / A : the observers measure the same acceleration for the particle. This is important, because two of Newton's three laws of motion are statements about the acceleration of an object. Such statements will be valid for both observers if they are moving relative to each other with constant velocity. Actually, there is a fourth relation, saying that the
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This note was uploaded on 10/19/2011 for the course PHYSICS 53L taught by Professor Mueller during the Spring '07 term at Duke.

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dynam1 - Physics 53 Dynamics 1 . from the same principles,...

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