Phys 325 Spring 2011 Lecture 4 - Physics 325 Lecture 4...

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Physics 325 Lecture 4 Solutions to Newton’s Equations of Motion (cont.) Finally, the most general case is: f)   ,, F F r v t In general, we cannot solve this analytically. But there are some special cases which can be solved analytically. If F can be factorized such that the variables separate, then things simplify considerably. We work in 1-D for simplicity and consider the three possible combinations of products of () hx , fv , and gt . ( i )       , F v t f v g t . Then         dv dv f v g t m g t dt m dt f v  And with a little luck, these integrals can be done. ( ii )       , F x v h x f v . Then     dv h x f v m dt Invoking again the chain rule dv dx dv dv v dt dt dx dx  gives         vdv h x f v mv h x dx m dx f v The integration yields   v x dx dt , and x ( t ) is then found from   dx dt vx ( iii )       , F x t h x g t . This is not solvable analytically unless either h or g is a constant. Instead, numerical techniques must be applied.
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Let’s now work two examples that exemplify the application of Newton’s Laws in solving for the motion of a particle under the influence of a force. Motion of a Charged Particle in a Uniform Magnetic Field Consider a particle of charge q moving in a uniform, constant magnetic field B that points in the z direction as shown: The net force on the particle is the magnetic force () F F v given by F qv B  The equation of motion is given by mv qv B (4.1) Note that equation (4.1) is really a set of three equations, one for each of the Cartesian components .Since ( , , ) (0,0, ) ( , ,0) x y z y x v v v v B B v B v B v B we have the components of (4.1): xy mv qBv (4.2) yx mv qBv  (4.3) 0 z mv (4.4) Equation (4.4) simply says that z v is constant, which makes sense because the magnetic force is always perpendicular to B . To simplify equations (4.2) and (4.3), lets define a parameter such that qB m which has units of inverse time and is called the cyclotron frequency .
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Using this definition, equations (4.2) and (4.3) become xy yx vv  (4.5)
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Phys 325 Spring 2011 Lecture 4 - Physics 325 Lecture 4...

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