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 acceleration due to gravity (g) is affected by changes in the radius and density of the Earth. Here’s a step-by-step breakdown:

### Step 1: Understand the formula for gravitational acceleration
The acceleration due to gravity (g) at the surface of a planet is given by the formula:
\[ g = \frac{GM}{R^2} \]
where:
- \( G \) is the gravitational constant,
- \( M \) is the mass of the Earth,
- \( R \) is the radius of the Earth.

### Step 2: Relate mass to density
The mass \( M \) of the Earth can also be expressed in terms of its density \( \rho \) and volume \( V \):
\[ M = \rho V \]
The volume \( V \) of a sphere (the Earth) is given by:
\[ V = \frac{4}{3} \pi R^3 \]
Thus, we can rewrite the mass as:
\[ M = \rho \left( \frac{4}{3} \pi R^3 \right) \]

### Step 3: Substitute mass in the gravitational formula
Substituting the expression for mass into the formula for \( g \):
\[ g = \frac{G \left( \rho \frac{4}{3} \pi R^3 \right)}{R^2} \]
This simplifies to:
\[ g = \frac{4}{3} \pi G \rho R \]

### Step 4: Analyze the new scenario
Now, if the radius of the Earth is increased by a factor of 2, we have:
\[ R' = 2R \]
We need to find the new density \( \rho' \) such that \( g' = g \).

### Step 5: Write the new gravitational acceleration
Substituting \( R' \) into the formula for \( g \):
\[ g' = \frac{4}{3} \pi G \rho' R' \]
Replacing \( R' \) with \( 2R \):
\[ g' = \frac{4}{3} \pi G \rho' (2R) \]
This simplifies to:
\[ g' = \frac{8}{3} \pi G \rho' R \]

### Step 6: Set the two expressions for g equal
To keep \( g' = g \):
\[ \frac{4}{3} \pi G \rho R = \frac{8}{3} \pi G \rho' R \]

### Step 7: Cancel common terms
Canceling \( \frac{4}{3} \pi G R \) from both sides gives:
\[ \rho = 2 \rho' \]

### Step 8: Solve for the new density
Rearranging gives us:
\[ \rho' = \frac{\rho}{2} \]

### Conclusion
Thus, the density must be changed by a factor of \( \frac{1}{2} \) to keep \( g \) the same when the radius is increased by a factor of 2.