Physics 325 Lecture 7 - Physics 325 Lecture 7 Newtons Laws...

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Physics 325 Lecture 7 Newton’s Laws in Three Dimensions We now turn to a general discussion of the motion of a particle in three dimensions. We already know the basic formula: dp F dt = G G (7.1) In three dimensions, this vector equation represents three decoupled equations: ( ) ( () ,,, , , , xx y z yx y z zx y z Fxy z vvvt m x z y z z = = = ±± ) (7.2) If the forces separate in a convenient fashion, such that F x depends only on x,v x and t etc., then the solving these decoupled equations amounts to solving three one-dimensional problems. Generally speaking that doesn’t happen and the solution can become quite complicated. One of the ways to handle this difficulty is to make use of conservation laws, which often provide a great simplification. We have already used this technique in one dimension where we invented a potential energy function for a force F(x) ( ) Ux F d x dU Fx dx =− We went on from there to show that the total energy (kinetic+potential) was a constant. Conservation of total energy is a powerful problem solving concept. In three dimensions, we define the potential function through ( ) ( ) ,, Fxyz Uxyz =−∇ G G (7.3) (note that since the force is a vector function we have to use a derivative of the potential that returns a vector. This is the gradient operator.) It is apparent from Equation (7.3) that this definition implies that 0 FU ×= × ∇= G GGG So the definition (7.3) is only possible if the curl of the force is zero. There are forces that have non-zero curls. For such forces one cannot define a potential function. These
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are non-conservative forces. Forces for which we can define a potential function are conservative forces and obey conservation of kinetic+potential energy. Here’s an example of a non-conservative force: ( ) ˆˆ F A yi xj =−+ G . The curl of is F G () ˆ 2 kA x y Ak xy ⎛⎞ ∂∂ −−= ⎜⎟ ⎝⎠ ˆ . This force has a simple form in cylindrical coordinates: ( ) ˆ ˆ ˆ sin cos F A yi xj Ar i j Ar φ φφ =−+ = − + = G
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This note was uploaded on 10/06/2011 for the course PHYS 325 taught by Professor Staff during the Fall '08 term at University of Illinois, Urbana Champaign.

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Physics 325 Lecture 7 - Physics 325 Lecture 7 Newtons Laws...

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