Suppose the gravitational force varies inversely as the nth...

Suppose the gravitational force varies inversely as the nth power of distance the time period of planet in circular orbit of radius $\mathrm{R}$ around the sun will be proportional to (A) $\mathrm{R}^{[(\mathrm{n}+1) / 2]}$ (B) $\mathrm{R}^{[(\mathrm{n}-1) / 2]}$ (C) $\mathrm{R}^{\mathrm{n}}$ (D) $\mathrm{R}^{[(\mathrm{n}-1) / 2]}$

Suppose the gravitational force varies inversely as the nth power of distance the time period of planet in circular orbit of radius $\mathrm{R}$ around the sun will be proportional to
(A) $\mathrm{R}^{[(\mathrm{n}+1) / 2]}$
(B) $\mathrm{R}^{[(\mathrm{n}-1) / 2]}$
(C) $\mathrm{R}^{\mathrm{n}}$
(D) $\mathrm{R}^{[(\mathrm{n}-1) / 2]}$

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Key Concepts

Inverse Power Law

This concept refers to a relationship where the force between two masses decreases as a function of the distance raised to a certain power. In physics, especially in gravity, if the force varies as 1/r^n, then n determines the rate at which the force diminishes with increasing separation between the masses.

Centripetal Force in Circular Motion

In circular orbital motion, an inward force is required to keep the object moving along a circular path. This required centripetal force is provided by the gravitational attraction, and setting the gravitational force equal to the centripetal force is key to deriving relationships between orbital parameters like period and radius.

Proportional Reasoning and Dimensional Analysis

This method involves comparing the dimensions and the proportional relationships in the equations governing a system. In the context of orbital motion under an inverse power law, equating the gravitational force to the centripetal force allows one to solve for the orbital period as a function of radius, leading to a general expression for the dependence on R.

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