If the radius of the earth were increased by a factor of 2 keeping the mass constant, by what factor would its density have to be changed to keep g the same?

Options

A. `(1)/(8)`
B. 4
C. `(1)/(2)`
D. `(1)/(4)`

Answer

To solve the problem, we need to analyze how the gravitational acceleration \( g \) depends on the radius and density of the Earth.

### Step-by-Step Solution:

1. **Understanding the formula for gravitational acceleration**:
The formula for gravitational acceleration \( g \) at the surface of a spherical body is given by:
\[ g = \frac{GM}{r^2} \]
where \( G \) is the gravitational constant, \( M \) is the mass of the Earth, and \( r \) is the radius of the Earth.

2. **Relating mass and density**:
The mass \( M \) can be expressed in terms of density \( \rho \) and volume \( V \):
\[ M = \rho V \]
For a sphere, the volume \( V \) is given by:
\[ V = \frac{4}{3} \pi r^3 \]
Therefore, we can write:
\[ M = \rho \left(\frac{4}{3} \pi r^3\right) \]

3. **Substituting mass into the equation for \( g \)**:
Substituting the expression for \( M \) into the formula for \( g \):
\[ g = \frac{G \left(\rho \frac{4}{3} \pi r^3\right)}{r^2} \]
Simplifying this, we get:
\[ g = \frac{4}{3} G \pi \rho r \]
This shows that \( g \) is directly proportional to both the density \( \rho \) and the radius \( r \).

4. **Setting up the relationship to keep \( g \) constant**:
To keep \( g \) constant while changing the radius, we can set up the equation:
\[ r_1 \rho_1 = r_2 \rho_2 \]
where \( r_1 \) and \( \rho_1 \) are the initial radius and density, and \( r_2 \) and \( \rho_2 \) are the new radius and density.

5. **Given conditions**:
We are given that the radius is increased by a factor of 2:
\[ r_2 = 2r_1 \]

6. **Substituting into the equation**:
Substituting \( r_2 \) into the equation:
\[ r_1 \rho_1 = (2r_1) \rho_2 \]
Dividing both sides by \( r_1 \) (assuming \( r_1 \neq 0 \)):
\[ \rho_1 = 2 \rho_2 \]

7. **Solving for \( \rho_2 \)**:
Rearranging gives:
\[ \rho_2 = \frac{\rho_1}{2} \]
This means the density must be halved to keep \( g \) constant when the radius is doubled.

### Final Answer:
The density of the Earth must be changed by a factor of \( \frac{1}{2} \) to keep \( g \) the same.