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. `4GM_(p)//D_(p)^(2)`
B. `GM_(p)m//D_(p)^(2)`
C. `GM_(p)//D_(p)^(2)`
D. `4GM_(p)m//D_(p)^(2)`

Answer

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: Determine the Radius of the Planet
The diameter \( D_0 \) of the planet is given. The radius \( R \) can be calculated as:
\[ R = \frac{D_0}{2} \]

### Step 2: Write the Gravitational Force Equation
The gravitational force \( F_g \) acting on the particle of mass \( m \) due to the planet can be expressed using Newton's law of gravitation:
\[ F_g = \frac{G M_0 m}{R^2} \]
where \( G \) is the universal gravitational constant.

### Step 3: Relate Gravitational Force to Acceleration
The acceleration \( g \) due to gravity at the surface of the planet can be defined as the gravitational force per unit mass of the particle:
\[ g = \frac{F_g}{m} \]
Substituting the expression for \( F_g \):
\[ g = \frac{G M_0 m}{R^2 m} \]
Here, the mass \( m \) cancels out:
\[ g = \frac{G M_0}{R^2} \]

### Step 4: Substitute the Radius in Terms of Diameter
Now, substitute \( R = \frac{D_0}{2} \) into the equation for \( g \):
\[ g = \frac{G M_0}{\left(\frac{D_0}{2}\right)^2} \]
This simplifies to:
\[ g = \frac{G M_0}{\frac{D_0^2}{4}} = \frac{4 G M_0}{D_0^2} \]

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

### Final Answer
The acceleration due to gravity is \( g = \frac{4 G M_0}{D_0^2} \).

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