In thinking about the uniform distribution of a gravitational field over a sphere, why does gravitational

Question

force have an inverse-square relationship with the distance between objects?

because the surface area of a sphere is proportional to its radius

because the volume of a sphere is proportional to its radius

because the surface area of a sphere is proportional to the square of its radius

because the volume of a sphere is proportional to the square of its radius

Answer 1

The reason gravitational force has an inverse-square relationship with the distance between objects is because the surface area of a sphere is proportional to the square of its radius.

Here's why:

1. According to Newton's law of universal gravitation, the force \( F \) between two masses \( m_1 \) and \( m_2 \) separated by a distance \( r \) is given by the formula: \[ F = G \frac{m_1 m_2}{r^2} \] where \( G \) is the gravitational constant.

2. The gravitational field strength (or gravitational force per unit mass) at a distance \( r \) from a mass \( M \) is also proportional to \( \frac{1}{r^2} \).

3. As the distance \( r \) increases, the gravitational force spreads out over the surface area of an imaginary sphere surrounding the mass \( M \). The surface area \( A \) of a sphere is given by the formula: \[ A = 4\pi r^2 \] This shows that the surface area increases with the square of the radius.

4. Therefore, as you move away from the mass, the gravitational force per unit area diminishes because the same amount of gravitational influence is distributed over a larger surface area, resulting in the inverse-square relationship.

In summary, the correct choice is:

• because the surface area of a sphere is proportional to the square of its radius.