### Correct Option: A. \( \frac{GM}{r^2} \) #### Detailed Explanation: 1. **Understanding Gravitational Field Strength**: - The gravitational field strength (\( g \)) at a point in space is defined as the force per unit mass experienced by a small test mass placed at that point. It is a vector quantity that points towards the mass creating the gravitational field. 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: \[ F = \frac{GMm}{r^2} \] - Here, \( G \) 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 \( g \) at a distance \( r \) from the mass \( M \), we can rearrange the formula for gravitational force: \[ g = \frac{F}{m} \] - Substituting the expression for \( F \): \[ g = \frac{GMm/r^2}{m} \] - The mass \( m \) cancels out: \[ g = \frac{GM}{r^2} \] - This shows that the gravitational field strength at a distance \( r \) from a point mass \( M \) is indeed \( \frac{GM}{r^2} \). 4. **Why Option A is Correct**: - Option A correctly represents the derived formula for gravitational field strength, confirming that it is directly proportional to the mass \( M \) and inversely proportional to the square of the distance \( r \). #### Explanation of Other Options: - **Option B: \( \frac{G}{Mr^2} \)** - This option incorrectly places \( M \) in the denominator. The gravitational field strength should be proportional to the mass \( M \), not inversely. Thus, this option is incorrect. - **Option C: \( \frac{M}{Gr^2} \)** - Similar to option B, this option incorrectly suggests that \( G \) is in the denominator. The gravitational field strength should not have \( G \) in the denominator, making this option incorrect. - **Option D: \( \frac{Gr^2}{M} \)** - This option also misrepresents the relationship. It suggests that gravitational field strength increases with \( r^2 \) and decreases with \( M \), which contradicts the established relationship. Therefore, this option is incorrect. ### Summary of Key Points: - 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 creating the field and inversely proportional to the square of the distance from the mass. - Understanding the correct relationships in the formula is crucial for solving problems related to gravitational fields.