Assume that earth is a spherical planet of uniform density `rho` , radius `R_e` mass `M_e` and acceleration due to gravity g. Then the gravitational constant G can be written as
a) `(3g)/(4pirhoR_e)` b) `(gR_e^2)/(M_e)` c) `(3g)/(4piR_e^2)` d) `(12g)/(4pirhoR_2)`

Options

A. only a & b are true
B. only b & c are true
C. only a, b &d are true
D. only a & d are true

Answer

To find the expression for the gravitational constant \( G \) in terms of the given parameters (density \( \rho \), radius \( R_e \), mass \( M_e \), and acceleration due to gravity \( g \)), we can follow these steps:

### Step 1: Write the formula for acceleration due to gravity
The formula for the acceleration due to gravity \( g \) at the surface of a spherical body is given by:

\[ g = \frac{G M_e}{R_e^2} \]

### Step 2: Express mass \( M_e \) in terms of density \( \rho \)
The mass \( M_e \) of the Earth can be expressed in terms of its density \( \rho \) and volume \( V \). The volume \( V \) of a sphere is given by:

\[ V = \frac{4}{3} \pi R_e^3 \]

Thus, the mass can be written as:

\[ M_e = \rho V = \rho \left( \frac{4}{3} \pi R_e^3 \right) \]

### Step 3: Substitute \( M_e \) back into the equation for \( g \)
Substituting the expression for \( M_e \) into the equation for \( g \):

\[ g = \frac{G \left( \rho \frac{4}{3} \pi R_e^3 \right)}{R_e^2} \]

### Step 4: Simplify the equation
Now, simplify the equation:

\[ g = \frac{G \rho \frac{4}{3} \pi R_e^3}{R_e^2} \]

This simplifies to:

\[ g = \frac{4}{3} G \rho \cdot R_e \]

### Step 5: Solve for \( G \)
Rearranging the equation to solve for \( G \):

\[ G = \frac{3g}{4 \pi \rho R_e} \]

### Conclusion
Thus, the gravitational constant \( G \) can be expressed as:

\[ G = \frac{3g}{4 \pi \rho R_e} \]

This matches option (a).