The equation provided is Newton's law of universal gravitation:

$F = G \frac{mM}{r^2}$

To solve for $m$, we need to isolate $m$ on one side of the equation.

Step 1: Isolate $m$

Multiply both sides of the equation by $r^2$ to remove the fraction:

$F \cdot r^2 = G \cdot mM$

Step 2: Solve for $m$

Next, divide both sides by $GM$ to isolate $m$:

$m = \frac{F \cdot r^2}{G \cdot M}$

So, the expression for $m$ is:

$m = \frac{F \cdot r^2}{G \cdot M}$

This is the formula to calculate the mass $m$ based on the gravitational force $F$, the gravitational constant $G$, the mass $M$ of the other object, and the distance $r$ between the centers of the two masses.

Would you like any further explanation or have other questions?

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Here are five related questions you might explore:

1. What is the value of $G$ (the gravitational constant), and how does it affect the calculation of $m$?
2. How would the formula change if we were solving for $r$ instead of $m$?
3. Can you describe a physical scenario where this equation might be applied?
4. How does the gravitational force $F$ change if the distance $r$ is doubled?
5. What are the units of $m$ if $F$ is in newtons, $r$ in meters, $M$ in kilograms, and $G$ in $\text{N} \cdot \text{m}^2/\text{kg}^2$?

Tip: Understanding how to manipulate algebraic equations is crucial in physics, especially when dealing with formulas like Newton's law of universal gravitation.