Transcription of 4. Central Forces - University of Cambridge
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4. Central Forces - University of Cambridge
4. Central ForcesIn this section we will study the three-dimensional motion of a particle in a centralforce potential. Such a system obeys the equation of motionm x= rV(r)( )where the potential depends only onr=|x|.Sincebothgravitationalandelectro staticforces are of this form, solutions to this equation contain some of the most importantresults in classical first line of attack in solving ( ) defined asL=mx xWe already saw in angular momentum is conserved in a centralpotential. The proof is straightforward:dLdt=mx x= x rV=0where the final equality follows becauserVis parallel conservation of angular momentum has an important consequence: all motiontakes place in a plane.
to this relationship in Section 4.3.2 when we discuss Kepler’s laws of planetary motion. • E min <E<0: Here the 1d system sits in the dip, oscillating backwards and forwards between two points. Of course, since l 6=0,theparticlealsohasangular velocity in the plane. This describes an orbit in which the radial distance r depends on time.
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