Final answer:
The frequency of small radial oscillations around a circular orbit under an inverse square law force can be shown to be equal to the orbital frequency. This involves understanding the relationship between centripetal force and the effective restoring force in small oscillations, which leads to the equivalence of the angular frequencies. Therefore, the correct option is B.
Step-by-step explanation:
The question posted by the student primarily involves frequencies related to harmonic motion and orbital motion under an inverse square law force, such as gravity or electrostatic force. This topic is typically covered in an introductory college-level physics course or an AP Physics class. The student is asked to find the frequency of small radial oscillations about steady circular motion for an attractive inverse square law force and show that it equals the orbital frequency. This problem touches upon concepts like centripetal frequency, angular frequency, radial frequency, and harmonic frequency. To answer this question, we recall that for a particle in a circular orbit under an inverse square law force, the force necessary to provide the centripetal acceleration can be given by Newton's universal law of gravitation (or Coulomb's law for electrostatic forces). When this particle undergoes small oscillations radially, we find that the effective force restoring it back to equilibrium is still proportional to the inverse square of the distance from the center (just like the centripetal force). Using this, and by setting it equivalent to Hooke's law for small oscillations, we can derive an expression for the frequency of these radial oscillations and show that it is the same as the frequency of the circular orbit. Specifically, the angular frequency of the radial oscillations (ω) will be the same as that of the orbital motion because the central force provides the restoring force necessary for the harmonic motion.