If the gravitational force between two objects were...

If the gravitational force between two objects were proportional to $(1 / \mathrm{R})\left(\right.$ and not as $\left.1 / \mathrm{R}^{2}\right)$ where $\mathrm{R}$ is separation between them, then a particle in circular orbit under such a force would have its orbital speed v proportional to (A) $\left(1 / \mathrm{R}^{2}\right)$ (B) $\mathrm{R}^{0}$ (C) $\mathrm{R}^{1}$ (D) $(1 / \mathrm{R})$

If the gravitational force between two objects were proportional to $(1 / \mathrm{R})\left(\right.$ and not as $\left.1 / \mathrm{R}^{2}\right)$ where $\mathrm{R}$ is separation between them, then a particle in circular orbit under such a force would have its orbital speed v proportional to
(A) $\left(1 / \mathrm{R}^{2}\right)$
(B) $\mathrm{R}^{0}$
(C) $\mathrm{R}^{1}$
(D) $(1 / \mathrm{R})$

Key Concepts

Orbital Velocity Derivation

The derivation of orbital velocity under a central force involves setting the centripetal force equal to the force of attraction. Under a force that varies as 1/R, this equality leads to a result where the orbital speed does not depend on the radius of the orbit. Understanding this derivation helps illustrate how different force laws lead to distinct orbital behaviors and emphasizes the importance of the force’s dependence on distance in central force problems.

Gravitational Force Laws

Gravitational force laws describe how the magnitude of the force between two masses depends on their separation. Traditionally, Newton’s law of universal gravitation states that the force falls off as the inverse square of the distance (1/R²). However, considering a modified law where the force is proportional to 1/R explores how different distance dependencies can fundamentally change the dynamics of objects under gravity.

Centripetal Force Requirement in Circular Orbits

In circular motion, an object must experience a centripetal force that is directed towards the center of the circular path to maintain its orbit. This required centripetal force is provided by the gravitational (or central) force acting on the object. By equating the centripetal force needed (mv²/R) with the force provided by the gravitational interaction (proportional to 1/R in the modified law), one can derive relationships between the orbital speed and the radius.

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