Problem 92 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

\( r = \sqrt{\frac{m M G}{F}} \)

Step 1: Isolate the Fraction

Start with the given equation: \[ F=G \frac{m M}{r^{2}} \]To isolate the fraction on one side, divide both sides by \( G \), resulting in:\[ \frac{F}{G} = \frac{m M}{r^{2}} \]

Step 2: Rearrange to Solve for \( r^2 \)

Multiply both sides by \( r^2 \) to move it to the numerator on the left side:\[ r^2 \times \frac{F}{G} = m M \]Next, solve for \( r^2 \) by dividing both sides by \( \frac{F}{G} \):\[ r^2 = \frac{m M}{\frac{F}{G}} = \frac{m M G}{F} \]

Step 3: Take the Square Root

To solve for \( r \), take the square root of both sides:\[ r = \sqrt{\frac{m M G}{F}} \]This expression gives the solution for \( r \) in terms of the other variables \( F, G, m, \) and \( M \).

Key Concepts

Variable Isolation

When dealing with algebraic equations, one of the main steps is isolating the variable of interest. In the equation given, we are tasked with solving for the variable \( r \). Variable isolation involves rearranging the equation so our variable of interest, here \( r \), stands alone on one side. This process often involves manipulating the equation to reverse any operations acting on the variable.

In our original equation \( F = G \frac{mM}{r^2} \), the fraction \( \frac{mM}{r^2} \) is multiplied by \( G \), and we want \( r \) by itself. The logical first step is to divide both sides of the equation by \( G \) to remove it from the right side. This allows us to focus on simplifying the relationship between the variables \( m \), \( M \), and \( r \). The result of this step is an equation that brings the fraction closer to isolation:

• \( \frac{F}{G} = \frac{mM}{r^2} \)

This process of variable isolation is crucial for further operations, such as solving for \( r^2 \), and eventually \( r \).

Square Root

Taking a square root is a common technique used when solving equations that include a squared term. Once we have isolated \( r^2 \) on one side, we need to undo the squaring to solve for \( r \). As a reminder, the square root operation will reverse the effect of squaring a number or expression.

In our specific equation, after isolating \( r^2 \), we arrive at:

• \( r^2 = \frac{mMG}{F} \)

To find \( r \), we take the square root of both sides. This will give us:

• \( r = \sqrt{\frac{mMG}{F}} \)

Remember, when solving square root equations, consider both positive and negative roots as potential solutions unless the context, such as a real-world scenario, dictates otherwise. Here, we assume \( r \) is a positive real number.

Fraction Manipulation

Fractions in algebraic equations can often seem daunting, but they can typically be handled with steady and consistent manipulation methods. When manipulating fractions, the goal is to simplify the equation to make it easier to solve for the variable of interest by getting rid of the fraction or rearranging it.

In the solution process, we see initial steps where the equation is divided and multiplied by terms to manipulate the given fraction:

• \( \frac{F}{G} = \frac{mM}{r^2} \)
• Next, to get \( r^2 \) on its own, multiply both sides by \( r^2 \), which moves \( r^2 \) to the numerator, simplifying the fraction further.
• Then, divide the right side by \( \frac{F}{G} \) to bring all terms in terms of \( mM, G, \) and \( F \).

It's important to apply consistent methods such as clearing the fractions via multiplication or division, ensuring the variable moves from the denominator. Following these approaches logically results in a well-defined expression for the variable of interest.