Question

4 multiple choice 1 point two planets are a distance of d-meters apart. they have a gravitational force of f newtons exerted on both of them. if the distance between them is decreased to d/2, what must the force be between them now in terms of the original force f? options: f/4, f/2, f, 2f, 4f

Explanation:

Step1: Recall Gravitational Force Formula

The gravitational force between two objects is given by Newton's law of gravitation: \( F = G\frac{m_1m_2}{d^2} \), where \( G \) is the gravitational constant, \( m_1 \) and \( m_2 \) are the masses of the two objects, and \( d \) is the distance between them.

Step2: Analyze the Change in Distance

Initially, the distance is \( d \), so the force is \( F = G\frac{m_1m_2}{d^2} \). Now, the distance is decreased to \( \frac{d}{2} \). Let the new force be \( F' \). Then, \( F' = G\frac{m_1m_2}{(\frac{d}{2})^2} \).

Step3: Simplify the Expression for \( F' \)

Simplify the denominator: \( (\frac{d}{2})^2 = \frac{d^2}{4} \). So, \( F' = G\frac{m_1m_2}{\frac{d^2}{4}} = 4G\frac{m_1m_2}{d^2} \). But from the initial formula, \( G\frac{m_1m_2}{d^2} = F \). Therefore, \( F' = 4F \).

Answer:

4F (the option corresponding to 4F)