### Correct Option: A. \( \frac{GM}{r^2} \) #### Detailed Explanation: 1. **Understanding Gravitational Field Strength**: - The gravitational field strength \( g \) at a distance \( r \) from a point mass \( M \) is defined as the force per unit mass experienced by a small test mass placed in the field. Mathematically, it can be expressed as: \[ g = \frac{F}{m} \] where \( F \) is the gravitational force acting on the test mass \( m \). 2. **Newton's Law of Universal Gravitation**: - According to Newton's law, the gravitational force \( F \) between two masses \( M \) (the point mass) and \( m \) (the test mass) separated by a distance \( r \) is given by: \[ F = \frac{GMm}{r^2} \] where \( G \) is the gravitational constant, approximately \( 6.674 \times 10^{-11} \, \text{N m}^2/\text{kg}^2 \). 3. **Deriving Gravitational Field Strength**: - To find the gravitational field strength \( g \), we substitute the expression for \( F \) into the equation for \( g \): \[ g = \frac{F}{m} = \frac{\frac{GMm}{r^2}}{m} \] - The mass \( m \) cancels out: \[ g = \frac{GM}{r^2} \] - This shows that the gravitational field strength \( g \) at a distance \( r \) from a point mass \( M \) is directly proportional to the mass \( M \) and inversely proportional to the square of the distance \( r \). 4. **Why Option A is Correct**: - Option A correctly represents the derived formula for gravitational field strength, \( g = \frac{GM}{r^2} \). It includes both the gravitational constant \( G \) and the mass \( M \), and it correctly shows the inverse square relationship with distance \( r \). #### Why the Other Options are Incorrect: - **Option B: \( \frac{G}{r^2} \)** - This option omits the mass \( M \). The gravitational field strength depends on the mass creating the field. Without \( M \), this expression does not represent the gravitational field strength due to a point mass. - **Option C: \( \frac{Mr^2}{G} \)** - This option incorrectly suggests that gravitational field strength increases with the square of the distance \( r \) and is inversely proportional to \( G \). This is not consistent with the established inverse square law of gravitation. - **Option D: \( \frac{M}{Gr^2} \)** - Similar to option C, this expression incorrectly places \( G \) in the denominator, suggesting that gravitational field strength decreases with increasing \( G \) and increases with \( r^2 \). This is contrary to the principles of gravitational attraction. ### Summary: - The gravitational field strength \( g \) at a distance \( r \) from a point mass \( M \) is given by \( g = \frac{GM}{r^2} \). - This formula is derived from Newton's law of universal gravitation. - Gravitational field strength is directly proportional to the mass \( M \) and inversely proportional to the square of the distance \( r \). - Options B, C, and D do not correctly represent the relationship defined by gravitational theory.