Problem 84 - Algebra and Trigonometry

Question

Solve the equation for the indicated variable. \(F=G \frac{m M}{r^{2}} ; \quad\) for \(m\)

Step-by-Step Solution

Answer

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

Step 1: Understand the Goal

The given equation is Newton's Law of Universal Gravitation: \(F = G \frac{mM}{r^2}\). Our goal is to solve this equation for the variable \(m\).

Step 2: Isolate the Target Variable

To solve for \(m\), we need to isolate \(m\) on one side of the equation. We start by multiplying both sides by \(r^2\) to eliminate the denominator:\( F \cdot r^2 = G \cdot m \cdot M \).

Step 3: Solve for the Variable

Now, solve for \(m\) by dividing both sides of the equation by \(G \cdot M\):\( m = \frac{F \cdot r^2}{G \cdot M} \).

Key Concepts

Variable Isolation

When dealing with equations in physics, isolating the target variable is often the first step. This means rearranging the equation so that the variable you want to find stands alone on one side. In our exercise, we were given the equation:\[ F = G \frac{mM}{r^2} \]Our aim was to solve for the variable \( m \). To isolate \( m \), we needed to perform a series of steps to keep it on one side of the equation, separate from other variables and constants. The strategy for isolation often involves:

• Eliminating fractions by multiplying each side by the denominator.
• Moving terms to different sides of the equation by performing inverse operations, like subtraction if a term is added.

In this example, we multiplied both sides by \( r^2 \) to start the process of isolating \( m \). This step was crucial to clear the denominator and set the stage for further simplification.

Solving Equations

Once we have isolated the variable we want to solve for, we move on to solving the equation completely. Solving an equation means performing the required mathematical operations to express the isolated variable purely in terms of other known quantities. Let's revisit our expression after clearing the fraction:\[ F \cdot r^2 = G \cdot m \cdot M \]Our task was to find \( m \), so we needed to rearrange this expression to have \( m \) alone on one side. This generally involves the use of operations that simplify the equation further.

• We divided both sides of the equation by \( G \cdot M \) to solve for \( m \).
• This resulted in the expression: \( m = \frac{F \cdot r^2}{G \cdot M} \).

These operations allow us to achieve a form where \( m \) is expressed in terms of other variables, and we can easily substitute numerical values if needed.

Mathematical Manipulation

Mathematical manipulation is a fundamental skill in solving equations and involves applying algebraic principles to change the form of an expression or equation. In the context of our exercise on Newton's Law of Universal Gravitation, mathematical manipulation helped us reach the final expression for \( m \), which was our goal.Our journey from \( F = G \frac{mM}{r^2} \) to \( m = \frac{F \cdot r^2}{G \cdot M} \) involved several manipulation techniques:

• Clearing fractions by multiplying through by the least common denominator.
• Systematically performing inverse operations to transfer terms from one side of the equation to another.
• Balancing the equation by applying the same operation to both sides.

Each step ensured that while transformations were applied, the equality and balance of the equation were maintained. Mastery of these techniques can assist students in tackling even more complicated equations with confidence and ease.