A spherical planet far out in space has a mass `M_(0)` and diameter `D_(0)`. A particle of mass m falling freely near the surface of this planet will experience an accelertion due to gravity which is equal to

Options

A. `GM_(0)//D_(0)^(2)`
B. `4mGM_(0)//D_(0)^(2)`
C. `4GM_(0)//D_(0)^(2)`
D. `GmM_(0)//D_(0)^(2)`

Explanation

To find the acceleration due to gravity experienced by a particle of mass \( m \) falling freely near the surface of a spherical planet with mass \( M_0 \) and diameter \( D_0 \), we can follow these steps:

### Step 1: Understand the gravitational force formula
The gravitational force \( F \) between two masses \( M_0 \) (the mass of the planet) and \( m \) (the mass of the particle) is given by Newton's law of gravitation:
\[ F = \frac{G M_0 m}{r^2} \]
where \( G \) is the gravitational constant and \( r \) is the distance between the centers of the two masses.

### Step 2: Determine the radius of the planet
The radius \( r \) of the planet can be calculated from its diameter \( D_0 \):
\[ r = \frac{D_0}{2} \]

### Step 3: Substitute the radius into the gravitational force formula
Substituting \( r = \frac{D_0}{2} \) into the gravitational force formula gives:
\[ F = \frac{G M_0 m}{\left(\frac{D_0}{2}\right)^2} \]
This simplifies to:
\[ F = \frac{G M_0 m}{\frac{D_0^2}{4}} = \frac{4 G M_0 m}{D_0^2} \]

### Step 4: Calculate the acceleration due to gravity
The acceleration \( g \) due to gravity experienced by the mass \( m \) is given by the formula:
\[ g = \frac{F}{m} \]
Substituting the expression for \( F \) from the previous step:
\[ g = \frac{4 G M_0 m}{D_0^2} \cdot \frac{1}{m} = \frac{4 G M_0}{D_0^2} \]

### Step 5: Finalize the expression for acceleration due to gravity
Thus, the acceleration due to gravity \( g \) experienced by the particle of mass \( m \) near the surface of the planet is:
\[ g = \frac{4 G M_0}{D_0^2} \]