A spherical planet has a mass MP and diameter DP A particle of mass m falling freely near the su

step 1 A spherical planet has a mass mp, mass has mp, and the radius of this planet is diameter is dp a

A spherical planet has a mass MP and diameter DP A particle...

A spherical planet has a mass $\mathrm{M}_{\mathrm{P}}$ and diameter $\mathrm{D}_{\mathrm{P}} \mathrm{A}$ particle of mass $\mathrm{m}$ falling freely near the surface of this planet will experience an acceleration due to gravity, equal to (A) $\left[\left(4 \mathrm{GM}_{\mathrm{P}}\right) /\left(\mathrm{D}_{\mathrm{P}}^{2}\right)\right]$ (B) $\left[\left(\mathrm{GM}_{\mathrm{P}} \mathrm{m}\right) /\left(\mathrm{D}_{\mathrm{P}}^{2}\right)\right]$ (C) $\left[\left(\mathrm{GM}_{\mathrm{P}}\right) /\left(\mathrm{D}_{\mathrm{P}}^{2}\right)\right]$ (D) $\left[\left(4 \mathrm{GM}_{\mathrm{P}} \mathrm{m}\right) /\left(\mathrm{D}_{\mathrm{P}}^{2}\right)\right]$

A spherical planet has a mass $\mathrm{M}_{\mathrm{P}}$ and diameter $\mathrm{D}_{\mathrm{P}} \mathrm{A}$ particle of mass $\mathrm{m}$ falling freely near the surface of this planet will experience an acceleration due to gravity, equal to
(A) $\left[\left(4 \mathrm{GM}_{\mathrm{P}}\right) /\left(\mathrm{D}_{\mathrm{P}}^{2}\right)\right]$
(B) $\left[\left(\mathrm{GM}_{\mathrm{P}} \mathrm{m}\right) /\left(\mathrm{D}_{\mathrm{P}}^{2}\right)\right]$
(C) $\left[\left(\mathrm{GM}_{\mathrm{P}}\right) /\left(\mathrm{D}_{\mathrm{P}}^{2}\right)\right]$
(D) $\left[\left(4 \mathrm{GM}_{\mathrm{P}} \mathrm{m}\right) /\left(\mathrm{D}_{\mathrm{P}}^{2}\right)\right]$

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Key Concepts

Relationship Between Diameter and Radius

For a sphere, the diameter is twice the radius. When the gravitational acceleration formula g = GM/R² is provided, if the problem statement gives the diameter instead of the radius, it is necessary to substitute R with half of the diameter. This conversion is critical to correctly express the gravitational acceleration in terms of the provided geometric dimensions.

Newton's Law of Universal Gravitation

This law states that every point mass attracts every other point mass in the universe with a force that is proportional to the product of their masses and inversely proportional to the square of the distance between them. It is the fundamental principle behind the calculation of gravitational forces between objects, such as a planet and a particle near its surface.

Gravitational Acceleration

Gravitational acceleration is the acceleration experienced by an object due to the gravitational force of a massive body, such as a planet. It is obtained by dividing the gravitational force by the mass of the object, resulting in an expression that depends solely on the mass of the planet and the distance from its center (typically given by g = GM/R²).

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