### Correct Option: A. \( \frac{GM}{r^2} \) #### Explanation of the Correct Answer According to Newton's law of universal gravitation, the strength of the gravitational field (also known as gravitational field strength or gravitational acceleration) at a distance \( r \) from a mass \( M \) is given by the formula: \[ g = \frac{GM}{r^2} \] Where: - \( g \) is the gravitational field strength (measured in Newtons per kilogram, N/kg), - \( G \) is the universal gravitational constant, approximately \( 6.674 \times 10^{-11} \, \text{N m}^2/\text{kg}^2 \), - \( M \) is the mass creating the gravitational field (in kilograms), - \( r \) is the distance from the center of the mass to the point where the gravitational field strength is being calculated (in meters). **Step-by-Step Explanation:** 1. **Understanding Gravitational Field Strength**: The gravitational field strength at a point in space is defined as the force experienced by a unit mass placed at that point. It tells us how strong the gravitational pull is at that location. 2. **Newton's Law of Universal Gravitation**: This law states that every mass attracts every other mass with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers. The formula for the gravitational force \( F \) between two masses \( m_1 \) and \( m_2 \) separated by a distance \( r \) is: \[ F = \frac{G m_1 m_2}{r^2} \] 3. **Deriving Gravitational Field Strength**: If we consider one of the masses (let's say \( M \)) to be much larger than the other (like a planet or star), we can define the gravitational field strength \( g \) at a distance \( r \) from \( M \) as the force per unit mass experienced by a small test mass \( m \): \[ g = \frac{F}{m} = \frac{GMm/r^2}{m} = \frac{GM}{r^2} \] This shows that the gravitational field strength depends only on the mass \( M \) and the distance \( r \), not on the mass of the test object. 4. **Units**: The units of \( g \) are derived from the formula. Since \( G \) has units of \( \text{N m}^2/\text{kg}^2 \) and \( r^2 \) has units of \( \text{m}^2 \), the units of \( g \) become: \[ \text{N/kg} = \frac{\text{N m}^2/\text{kg}^2}{\text{m}^2} = \text{N/kg} \] This confirms that \( g \) is indeed a measure of force per unit mass. #### Explanation of Incorrect Options - **Option B: \( \frac{G}{r^2} \)** This option is incorrect because it omits the mass \( M \). The gravitational field strength must depend on the mass creating the field. Without \( M \), this expression does not represent a gravitational field strength. - **Option C: \( \frac{M}{r^2} \)** This option is also incorrect for the same reason as option B. It lacks the gravitational constant \( G \), which is essential for converting the mass into a force per unit mass. The gravitational field strength cannot be expressed solely in terms of mass and distance without including \( G \). - **Option D: \( \frac{GM}{r} \)** This option is incorrect because it suggests a linear relationship with distance \( r \) rather than an inverse square relationship. The gravitational field strength decreases with the square of the distance, not linearly. ### Revision Summary - The gravitational field strength \( g \) at a distance \( r \) from a mass \( M \) is given by \( g = \frac{GM}{r^2} \). - The formula reflects the inverse square law of gravitation, indicating that gravitational strength decreases with the square of the distance. - The universal gravitational constant \( G \) is crucial for relating mass and distance in gravitational calculations. - Incorrect options either omit necessary components or misrepresent the relationship between gravitational strength and distance.