The correct option for the question regarding the gravitational field strength at a distance \( r \) from a point mass \( M \) is: **B. The gravitational field strength is directly proportional to the mass \( M \) and inversely proportional to the distance \( r^2 \).** ### Detailed Explanation 1. **Understanding Gravitational Field Strength**: - The gravitational field strength \( g \) at a distance \( r \) from a point mass \( M \) is defined as the force \( F \) experienced by a unit mass \( m \) placed at that distance. Mathematically, it is expressed as: \[ g = \frac{F}{m} \] - According to Newton's law of universal gravitation, the force \( F \) between two masses \( M \) and \( m \) is given by: \[ F = \frac{G M m}{r^2} \] where \( G \) is the gravitational constant. 2. **Deriving Gravitational Field Strength**: - Substituting the expression for \( F \) into the equation for \( g \): \[ g = \frac{F}{m} = \frac{G M m}{r^2 m} = \frac{G M}{r^2} \] - This shows that the gravitational field strength \( g \) is directly proportional to the mass \( M \) and inversely proportional to the square of the distance \( r \). 3. **Proportional Relationships**: - From the derived formula \( g = \frac{G M}{r^2} \): - **Directly Proportional to \( M \)**: If the mass \( M \) increases, the gravitational field strength \( g \) increases linearly. For example, if \( M \) doubles, \( g \) also doubles. - **Inversely Proportional to \( r^2 \)**: If the distance \( r \) increases, the gravitational field strength \( g \) decreases with the square of that distance. For instance, if \( r \) doubles, \( g \) becomes one-fourth of its original value. ### Analysis of Other Options - **Option A**: [blank] - This option does not provide any information, so it cannot be correct. - **Option C**: [blank] - Similar to Option A, this option lacks content and cannot be evaluated. - **Option D**: "The gravitational field strength is inversely proportional to the mass \( M \) and directly proportional to the distance \( r^2 \)." - This statement is incorrect. It suggests that as the mass \( M \) increases, the gravitational field strength \( g \) decreases, which contradicts the derived relationship. Additionally, it incorrectly states that \( g \) is directly proportional to \( r^2 \), which is also false. The correct relationship shows that \( g \) decreases as \( r^2 \) increases. ### Common Pitfalls - **Confusing Proportional Relationships**: Students often mix up direct and inverse relationships. Remember that "directly proportional" means that as one quantity increases, the other does too, while "inversely proportional" means that as one quantity increases, the other decreases. - **Forgetting the Square in Distance**: It's crucial to remember that gravitational field strength depends on the square of the distance, not just the distance itself. ### Revision Summary - The gravitational field strength \( g \) is given by \( g = \frac{G M}{r^2} \). - It is **directly proportional** to the mass \( M \) and **inversely proportional** to the square of the distance \( r \). - Always check the relationships carefully to avoid confusion between direct and inverse proportionality. - Remember that increasing the distance decreases the gravitational field strength significantly due to the square relationship.