The correct option for the gravitational field strength at a distance \( r \) from a point mass \( M \) in space is **A. \( \frac{GM}{r^2} \)**. ### Detailed Explanation 1. **Understanding Gravitational Field Strength**: - Gravitational field strength (\( g \)) is defined as the force per unit mass experienced by a small test mass placed in the field. It is a vector quantity that points towards the mass creating the gravitational field. - The formula for gravitational field strength due to a point mass \( M \) at a distance \( r \) is derived from Newton's law of universal gravitation. 2. **Newton's Law of Universal Gravitation**: - According to Newton's law, the gravitational force \( F \) between two masses \( M \) (the mass creating the field) 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 distance \( r \), we divide the gravitational force \( F \) by the test mass \( m \): \[ g = \frac{F}{m} = \frac{GMm/r^2}{m} = \frac{GM}{r^2} \] - This shows that the gravitational field strength \( g \) at a distance \( r \) from a point mass \( M \) is indeed \( \frac{GM}{r^2} \). ### Why the Other Options are Incorrect - **Option B: \( \frac{G}{r^2} \)**: - This expression lacks the mass \( M \) in the numerator. 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 implies that the gravitational field strength is proportional to the mass \( M \) alone, without considering the effects of the gravitational constant, which is essential for the correct calculation of gravitational interactions. - **Option D: \( \frac{GMr}{r^3} \)**: - This expression simplifies to \( \frac{GM}{r^2} \) when \( r \) is not zero, but it is unnecessarily complicated and misleading. It suggests a dependence on \( r \) in a way that is not standard for gravitational field strength, which is simply \( \frac{GM}{r^2} \). ### 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. - 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 gravitational calculations. By keeping these points in mind, you can confidently approach questions related to gravitational fields in your physics studies.