Phys 325 Spring 2011 Lecture 4

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

This preview shows pages 1–4. Sign up to view the full content.

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.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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 .
Using this definition, equations (4.2) and (4.3) become xy yx vv  (4.5)

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 7

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

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online