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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.

Learn how to solve for m using Newton's law of universal gravitation with a detailed step-by-step solution. This problem covers gravitational force calculations, including the use of the gravitational constant (G), mass (M), and distance (r). Suitable for advanced high school students.

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