One can easily "weigh the earth" by calculating the mass of earth using the formula (in usual notation)

Options

A. `G/g R_(E)^(2)`
B. `g/G R_(E)^(2)`
C. `g/G R_(E)`
D. `G/g R_(E)^(3)`

Answer

To calculate the mass of the Earth using the gravitational force formula, we can follow these steps:

### Step 1: Understand the Gravitational Force
The gravitational force \( F \) acting on a body of mass \( m \) near the Earth's surface is given by:
\[ F = mg \]
where \( g \) is the acceleration due to gravity.

### Step 2: Apply Newton's Law of Universal Gravitation
According to Newton's Law of Universal Gravitation, the gravitational force between two masses (in this case, the Earth and the body) can also be expressed as:
\[ F = \frac{GMm}{r^2} \]
where:
- \( G \) is the gravitational constant,
- \( M \) is the mass of the Earth,
- \( r \) is the radius of the Earth.

### Step 3: Set the Two Expressions for Force Equal
Since both expressions represent the same gravitational force, we can set them equal to each other:
\[ mg = \frac{GMm}{r^2} \]

### Step 4: Simplify the Equation
We can cancel \( m \) from both sides (assuming \( m \neq 0 \)):
\[ g = \frac{GM}{r^2} \]

### Step 5: Solve for the Mass of the Earth
Rearranging the equation to solve for \( M \) (the mass of the Earth), we get:
\[ M = \frac{gr^2}{G} \]

### Conclusion
Thus, the mass of the Earth can be calculated using the formula:
\[ M = \frac{gr^2}{G} \]
where \( g \) is the acceleration due to gravity, \( r \) is the radius of the Earth, and \( G \) is the gravitational constant.