### Correct Option: A. \( \frac{GM}{r^2} \) #### Detailed Explanation: 1. **Understanding Gravitational Field Strength**: - The gravitational field strength (\( g \)) at a point in space due to a mass \( M \) is defined as the force per unit mass experienced by a small test mass placed at that point. Mathematically, it is 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 \( g \) at a distance \( r \) from the mass \( M \), we substitute the expression for \( F \) into the equation for \( g \): \[ g = \frac{F}{m} = \frac{GMm/r^2}{m} \] - 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. **Understanding the Variables**: - \( G \) (gravitational constant) is a universal constant that quantifies the strength of gravity. - \( M \) is the mass creating the gravitational field. - \( r \) is the distance from the center of the mass \( M \) to the point where the gravitational field strength is being calculated. #### Why Other Options Are Incorrect: - **Option B: \( \frac{G}{r^2} \)**: - This expression lacks the mass \( M \). It suggests that the gravitational field strength depends only on the gravitational constant and the distance, which is incorrect. The strength of the gravitational field is directly proportional to the mass creating it. - **Option C: \( \frac{M}{r^2} \)**: - Similar to option B, this expression also omits the gravitational constant \( G \). It incorrectly implies that the gravitational field strength is proportional to the mass \( M \) alone, without considering the universal gravitational constant that relates mass to gravitational force. - **Option D: \( \frac{GM}{r} \)**: - This expression is incorrect because it suggests that gravitational field strength decreases linearly with distance \( r \), rather than with the square of the distance. The correct relationship shows that as you move away from the mass, the gravitational field strength decreases with the square of the distance. ### 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 relationship is derived from Newton's law of universal gravitation. - The gravitational field strength is proportional to the mass \( M \) and inversely proportional to the square of the distance \( r \). - Understanding the role of the gravitational constant \( G \) is crucial in calculating gravitational effects.