Mass of the earth has been determined through

Options

A. use of Kepler's `(T^(2))/(R^(3))` constancy law
B. sampling the density of earth's crust and using earth's radius
C. Cavandish's determination of G and using earth's radius and g at its surface
D. use of periods of satellite at different heights above earth's surface.

Explanation

To determine the mass of the Earth, we can use the following steps:

### Step 1: Understanding Gravitational Force
The gravitational force between two masses is given by Newton's law of gravitation:
\[ F = \frac{G \cdot m_1 \cdot m_2}{r^2} \]
where:
- \( F \) is the gravitational force,
- \( G \) is the gravitational constant,
- \( m_1 \) and \( m_2 \) are the masses of the two objects,
- \( r \) is the distance between the centers of the two masses.

### Step 2: Applying to Earth
In the case of the Earth, we can consider:
- \( m_1 \) as the mass of the Earth (\( M \)),
- \( m_2 \) as the mass of an object on the surface of the Earth (\( m \)),
- \( r \) as the radius of the Earth (\( R \)).

The gravitational force acting on the object can also be expressed as:
\[ F = m \cdot g \]
where \( g \) is the acceleration due to gravity at the Earth's surface.

### Step 3: Setting the Forces Equal
Since both expressions represent the same gravitational force, we can set them equal to each other:
\[ m \cdot g = \frac{G \cdot M \cdot m}{R^2} \]

### Step 4: Simplifying the Equation
We can cancel \( m \) from both sides (assuming \( m \neq 0 \)):
\[ g = \frac{G \cdot M}{R^2} \]

### Step 5: Solving for Mass of the Earth
Rearranging the equation to solve for \( M \):
\[ M = \frac{g \cdot R^2}{G} \]

### Step 6: Conclusion
Thus, the mass of the Earth can be determined using the value of \( g \) (acceleration due to gravity at the surface), the radius of the Earth \( R \), and the gravitational constant \( G \). This method is often referred to as the Cavendish determination of \( G \) and is a fundamental way to calculate the mass of the Earth.

### Final Answer
The mass of the Earth has been determined through the Cavendish determination of \( G \) and using Earth's radius.

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