The acceleration due to gravity at a piece depends on the mass of the earth M, radius of the earth R and the gravitational constant G. Derive an expression for g?

Explanation

To derive the expression for the acceleration due to gravity \( g \) at the surface of the Earth, we will follow these steps:

### Step 1: Understand the Gravitational Force
The gravitational force \( F \) acting on a mass \( m \) at the surface of the Earth can be expressed using Newton's law of universal gravitation:

\[ F = \frac{G \cdot M \cdot m}{R^2} \]

Where:
- \( G \) is the gravitational constant,
- \( M \) is the mass of the Earth,
- \( R \) is the radius of the Earth,
- \( m \) is the mass of the object experiencing the gravitational force.

### Step 2: Apply Newton's Second Law of Motion
According to Newton's second law of motion, the force acting on an object is also given by:

\[ F = m \cdot g \]

Where:
- \( g \) is the acceleration due to gravity.

### Step 3: Set the Two Expressions for Force Equal
Since both expressions represent the same force \( F \), we can set them equal to each other:

\[ m \cdot g = \frac{G \cdot M \cdot m}{R^2} \]

### Step 4: Cancel Out the Mass \( m \)
Assuming \( m \) is not zero, we can divide both sides of the equation by \( m \):

\[ g = \frac{G \cdot M}{R^2} \]

### Step 5: Final Expression
Thus, we have derived the expression for the acceleration due to gravity \( g \) at the surface of the Earth:

\[ g = \frac{G \cdot M}{R^2} \]

### Summary
The expression for the acceleration due to gravity at the surface of the Earth is given by:

\[ g = \frac{G \cdot M}{R^2} \]