An isotropic oscillator potential V(r)==kr leads, like the gravitational -1/r potential, to closed planar orbits. Suppose,...

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An isotropic oscillator potential V(r)==kr² leads, like the gravitational -1/r potential, to closed planar orbits. Suppose, however, that the "spring" has a non- zero relaxed length a, so that: V(r) = /k(r-a)² and consider the (planar) motions of a particle of mass m and initial angular momentum L in this potential. Find a condition relating the radius ro of circular orbits and the given constants. (It is not necessary to solve for ro.) (b) For nearly circular orbits, find the angular change between successive radial maxima. For a/ro < 1, express your answer to first order in a/ro (eliminate L from the answer). Check that the orbits are indeed closed for a/ro → 0 An isotropic oscillator potential V(r)==kr² leads, like the gravitational -1/r potential, to closed planar orbits. Suppose, however, that the "spring" has a non- zero relaxed length a, so that: V(r) = /k(r-a)² and consider the (planar) motions of a particle of mass m and initial angular momentum L in this potential. Find a condition relating the radius ro of circular orbits and the given constants. (It is not necessary to solve for ro.) (b) For nearly circular orbits, find the angular change between successive radial maxima. For a/ro < 1, express your answer to first order in a/ro (eliminate L from the answer). Check that the orbits are indeed closed for a/ro → 0