To solve for \( M \) in the equation \( F = \frac{G \cdot m \cdot M}{d^2} \), you can rearrange the formula as follows:

\[
M = \frac{F \cdot d^2}{G \cdot m}
\]

This isolates \( M \) on one side of the equation.

### Steps:
1. Multiply both sides of the equation by \( d^2 \) to get rid of the denominator:
   \[
   F \cdot d^2 = G \cdot m \cdot M
   \]
2. Divide both sides by \( G \cdot m \) to solve for \( M \):
   \[
   M = \frac{F \cdot d^2}{G \cdot m}
   \]

This gives the value of \( M \) in terms of \( F \), \( G \), \( m \), and \( d \).

Would you like further details or have any questions? 

Here are five related questions:
1. How does changing the distance \( d \) affect the value of \( M \)?
2. What is the significance of the gravitational constant \( G \) in this equation?
3. How can this formula be applied to determine the mass of a planet?
4. If the force \( F \) doubles, what happens to the mass \( M \)?
5. How would you solve for \( m \) instead of \( M \) in this equation?

**Tip:** When dealing with gravitational force equations, always ensure your units are consistent, especially when using the gravitational constant \( G \).

Learn how to solve for mass M in the gravitational force equation F = G(mM/d^2) with a detailed step-by-step solution. This problem covers key physics concepts, including gravitational force, Newton's Law of Universal Gravitation, and algebraic manipulation. Suitable for students in Grades 9-12.

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