### 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. **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 \( G \) and the distance \( r \), which is incorrect. The strength of the gravitational field is directly proportional to the mass creating the field. - **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 \( M \) without considering the universal gravitational constant, which is essential for the correct calculation of gravitational forces. - **Option D: \( \frac{GM}{r} \)**: - This expression is incorrect because it suggests that the gravitational field strength decreases linearly with distance \( r \), rather than with the square of the distance. The gravitational field strength decreases with the square of the distance from the mass, as shown in the derivation. #### Common Pitfalls: - **Confusing Gravitational Field Strength with Gravitational Force**: Remember that gravitational field strength is a measure of force per unit mass, while gravitational force is the total force acting between two masses. - **Neglecting the Gravitational Constant**: Always include \( G \) when calculating gravitational effects, as it is a fundamental constant that relates mass and distance in gravitational interactions. ### Revision 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 derives from Newton's law of universal gravitation. - Other options are incorrect due to missing components or incorrect relationships. - Always remember the role of the gravitational constant \( G \) in gravitational calculations.