The correct option for the gravitational field strength at a distance \( r \) from a point mass \( M \) 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) is given by: \[ F = \frac{G M m}{r^2} \] - Here, \( G \) is the gravitational constant, \( M \) is the mass creating the gravitational field, \( m \) is the mass experiencing the force, and \( r \) is the distance between the centers of the two masses. 3. **Deriving Gravitational Field Strength**: - To find the gravitational field strength \( g \) at a distance \( r \) from mass \( M \), we divide the gravitational force \( F \) by the test mass \( m \): \[ g = \frac{F}{m} = \frac{G M m}{r^2 m} = \frac{G M}{r^2} \] - 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: \( \frac{GM}{r} \)**: - This formula resembles the expression for gravitational potential, not gravitational field strength. Gravitational potential (\( V \)) at a distance \( r \) from a mass \( M \) is given by \( V = -\frac{GM}{r} \). Thus, this option is incorrect. - **Option C: \( \frac{GM^2}{r^2} \)**: - This expression incorrectly includes \( M^2 \) in the numerator. The gravitational field strength depends linearly on the mass \( M \) creating the field, not on \( M^2 \). Therefore, this option is also incorrect. - **Option D: \( \frac{G}{r^2} \)**: - This option omits the mass \( M \) entirely. The gravitational field strength is directly proportional to the mass creating the field. Without \( M \), this expression does not represent the gravitational field strength due to a point mass. Hence, 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 a measure of the force experienced by a unit mass in the gravitational field. - Other options either misrepresent the relationship between mass and distance or confuse gravitational field strength with gravitational potential. By understanding these concepts, you can confidently apply the formula for gravitational field strength in various physics problems.