### Correct Option: A. \( \frac{GM}{r^2} \) #### Detailed Explanation: To understand why option A is the correct answer, we need to delve into the concept of gravitational field strength and how it behaves outside a spherical mass, such as a planet. 1. **Definition of Gravitational Field Strength (g)**: The gravitational field strength at a point in space 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 and \( m \) is the mass of the test object. 2. **Newton's Law of Universal Gravitation**: According to Newton's law, the gravitational force \( F \) between two masses \( M \) (the mass of the planet) and \( m \) (the mass of the test object) separated by a distance \( r \) is given by: \[ F = \frac{GMm}{r^2} \] where \( G \) is the gravitational constant. 3. **Calculating Gravitational Field Strength**: To find the gravitational field strength \( g \) at a distance \( r \) from the center of the planet, we can substitute the expression for \( F \) into the definition of \( g \): \[ g = \frac{F}{m} = \frac{GMm}{r^2 \cdot m} \] The mass \( m \) of the test object cancels out: \[ g = \frac{GM}{r^2} \] This shows that the gravitational field strength at a distance \( r \) from the center of a planet of mass \( M \) is indeed \( \frac{GM}{r^2} \). 4. **Understanding the Context**: The condition that \( r \) is greater than \( R \) (the radius of the planet) is crucial because it ensures that we are outside the planet. Inside a planet, the gravitational field strength would vary depending on the distance from the center, but outside, it behaves as if all the mass were concentrated at a point at the center. #### Why Other Options Are Incorrect: - **Option B: \( \frac{GM}{R^2} \)**: This expression represents the gravitational field strength at the surface of the planet (where \( r = R \)). It does not apply when \( r \) is greater than \( R \), making it incorrect for our scenario. - **Option C: \( \frac{gR^2}{r^2} \)**: This option is incorrect because it introduces \( g \) (the gravitational field strength at the surface) in a way that does not relate to the gravitational field strength at distance \( r \). It does not follow from the principles of gravitational force and is not a standard expression. - **Option D: \( \frac{gM}{r} \)**: This option is also incorrect. It incorrectly combines \( g \) (which is dependent on \( R \) when at the surface) with \( M \) and \( r \) in a way that does not reflect the inverse square law of gravitation. The gravitational field strength does not depend linearly on \( r \) but rather inversely on \( r^2 \). ### Summary for Revision: - The gravitational field strength \( g \) at a distance \( r \) from a planet of mass \( M \) is given by \( g = \frac{GM}{r^2} \). - This formula is derived from Newton's law of universal gravitation. - The expression is valid only when \( r > R \), meaning we are outside the planet. - Other options either apply to different scenarios or misrepresent the relationship between gravitational force and distance.