Problem 69 - Algebra 2

Question

Solve each equation or formula for the specified variable. $$ F=G \frac{M m}{r^{2}}, \text { for } m $$

Step-by-Step Solution

Answer

The solution for \( m \) is \( m = \frac{F r^2}{G M} \).

Step 1: Identify the given formula

The given formula is:\[ F = G \frac{M m}{r^2} \]where \( F \) is the force, \( G \) is the universal gravitational constant, \( M \) and \( m \) are masses, and \( r \) is the distance between their centers. Our goal is to solve this equation for \( m \).

Step 2: Isolate the term containing the variable of interest

To isolate \( m \), we want to have \( \frac{M m}{r^2} \) by itself on one side of the equation. Start by multiplying both sides of the equation by \( \frac{r^2}{G} \) to clear the fraction:\[ F \cdot \frac{r^2}{G} = \frac{M m}{r^2} \cdot \frac{r^2}{G} \]

Step 3: Simplify the equation

After simplification, we have:\[ \frac{F r^2}{G} = M m \]The fraction \( \frac{r^2}{r^2} \) on the right side cancels out, leaving us with \( M m \).

Step 4: Solve for the specified variable

Now, to solve for \( m \), divide both sides of the equation by \( M \):\[ m = \frac{F r^2}{G M} \]

Step 5: Conclude the solution

We have successfully solved the formula for \( m \). The expression for \( m \) in terms of \( F, G, M, \text{and} \; r^2 \) is:\[ m = \frac{F r^2}{G M} \]

Key Concepts

Solving Equations

When we solve equations, we are essentially aiming to find the value of an unknown variable that makes the equation true. This process involves performing a series of logical and mathematical steps to isolate the variable we’re interested in. Let's consider the original exercise equation:

• The equation given is: \( F = G \frac{M m}{r^2} \)
• Our task is to solve for the variable \( m \).

The initial step is to identify the part of the equation containing \( m \) and work towards isolating it. This involves performing operations that simplify and rearrange the equation without altering its balance. By multiplying both sides by certain terms, canceling out fractions, or adding and subtracting similar terms, you can more clearly see the variable by itself. As we proceed with these steps, it's essential to apply the same operations to both sides of the equation to maintain equality. In this case, multiplying both sides by \( \frac{r^2}{G} \) starts the journey to isolating \( m \). This careful manipulation ensures that we solve equally on both sides of the equation.

Formula Rearrangement

Formula rearrangement is a fundamental skill in algebra often used when dealing with formulas involving multiple variables and constants. It's all about changing the order of the terms to solve for a specific variable. Rearranging formulas follows the same principles as solving equations, but with an emphasis on clarity and efficiency.
In our given problem, the formula rearrangement was used to solve for \( m \):

• Initially, the variable \( m \) is embedded within the fraction \( \frac{M m}{r^2} \).
• The steps to rearrange involve clearing the fraction by multiplying with its reciprocal, which in turn simplifies the arrangement to isolate \( m \).

The key here is to methodically undo the operations currently affecting \( m \). Multistep operations such as these often require understanding inverse operations: if something is being multiplied, you divide; if it’s added, you subtract, and so forth. Each rearrangement moves you closer to viewing the equation in its desired form. Here, the equation was expertly flipped and rearranged to let \( m \) stand alone by finally dividing everything else affecting \( m \).

Variables and Constants

In equations and formulas, it is crucial to differentiate between variables and constants. Variables are the symbols that can represent different values within a particular problem, whereas constants are fixed values. In the problem we solved, the variables included \( m \), \( F \), \( M \), and \( r \), while \( G \) represented the universal gravitational constant.
Understanding these distinctions helps in strategically solving the equation:

• \( m \) was our variable of interest—the quantity we wanted to isolate.
• The constant \( G \) played a stable role, allowing us to perform operations like multiplication or division without changing its value.

The clarity between what is constant and what is variable provides a stable framework to apply algebraic rules. Constants, due to their fixed nature, often end up as the divisors or multipliers that help simplify the problem or give scale to the equation. Grasping these concepts allows you to appropriately tackle the equation and address any algebraic challenge effectively.