Problem 88 - Algebra and Trigonometry

Question

\(79-92\) Solve the equation for the indicated variable. $$ F=G \frac{m M}{r^{2}} ; \quad \text { for } r $$

Step-by-Step Solution

Answer

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

Step 1: Understand the equation

The equation given is the formula for gravitational force: \( F = G \frac{mM}{r^2} \). We are required to solve this equation for the variable \( r \).

Step 2: Isolate the fraction

Start by isolating the fraction involving \( r \). This can be done by multiplying both sides of the equation by \( r^2 \): \[ F r^2 = G m M \]

Step 3: Solve for \( r^2 \)

We next isolate \( r^2 \) by dividing both sides by \( F \): \[ r^2 = \frac{G m M}{F} \]

Step 4: Solve for \( r \)

To solve for \( r \), we take the square root of both sides of the equation:\[ r = \sqrt{\frac{G m M}{F}} \]

Step 5: Consider the positive root

Since \( r \) represents a distance, we only consider the positive root: \[ r = \sqrt{\frac{G m M}{F}} \]

Key Concepts

Solving for a Variable

Solving for a variable involves rearranging an equation to isolate a specific variable. In our exercise, we are solving the formula for gravitational force to find \( r \). The original equation is:

• \( F = G \frac{mM}{r^2} \)

Here, \( F \) is the gravitational force, \( G \) is the gravitational constant, \( m \) and \( M \) are the masses, and \( r \) is the distance we want to find. To solve for \( r \), we need to rearrange the equation so that \( r \) stands alone on one side.
This process of isolating \( r \) will involve mathematical operations that work in reverse order of evaluation in equation hierarchy. First, focus on eliminating the terms that are directly combined with the variable using operations like multiplying or dividing. Ultimately, we aim to express \( r \) explicitly in terms of the other variables by removing all other terms from one side of the equation.

Manipulating Equations

Manipulating equations allows us to transform one form of an equation into another, helping to isolate a variable. In the given gravitational force equation, one necessary manipulation is multiplying both sides of the equation by \( r^2 \), which removes \( r^2 \) from the denominator:

• \( Fr^2 = GmM \)

With \( r^2 \) isolated, the next step is to solve for \( r^2 \) by dividing both sides by \( F \):

• \( r^2 = \frac{GmM}{F} \)

Through these steps, we've transformed the initial equation into a form where the target variable \( r^2 \) is isolated. It demonstrates how we can change the arrangement of terms while maintaining equality, a fundamental strategy when solving for a particular variable. Breaking down complex equations into manageable parts is key to mastering equation manipulation.

Square Root

To find the original variable from a squared term, like \( r^2 \), we use the square root. The operation of taking a square root undoes the squaring process:

• \( r = \sqrt{\frac{GmM}{F}} \)

The square root function acts as the inverse of the squaring operation, helping to obtain \( r \) from \( r^2 \).
It is important to consider that taking a square root yields both a positive and a negative value (since both \( x^2 \) and \( -x^2 \) equal \( x^2 \)). However, because \( r \) represents a real-world distance, only the positive root is physically meaningful:

• \( r = \sqrt{\frac{GmM}{F}} \)

Square roots can sometimes be intimidating, but understanding that they reverse squared values makes them invaluable in solving problems where squared variables are involved. This helps to collapse an equation to a simpler form, making it easier to work with real-world quantities.