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Chapter 7: Problem 76

Solve each formula for the specified variable \(r=\sqrt{\frac{M m}{F}}\) for \(F\)

Short Answer

Expert verified
F = \frac{M m}{r^2}

Step by step solution

01

- Square Both Sides

To isolate formula involving a square root, start by squaring both sides of the equation. This will eliminate the square root. So, square both sides of \( r = \sqrt{\frac{M m}{F}} \) to get: \( r^2 = \frac{M m}{F} \).
02

- Isolate F

Rearrange the equation to isolate F. Multiply both sides by F to get: \( F r^2 = M m \).
03

- Solve for F

Divide both sides by \( r^2 \) to solve for F: \( F = \frac{M m}{r^2} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Square Both Sides
When solving equations that involve square roots, the first step is often to square both sides of the equation. This step is crucial because it helps you eliminate the square root and work with simpler terms.

In our example, we start with the equation:

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

To remove the square root, we square both sides of the equation:

\[ r^2 = \frac{M m}{F} \]

Squaring both sides effectively cancels the square root, making it easier to work with the variable.
Isolate Variable
Next, we need to isolate the variable of interest. In this case, we want to solve for the variable \( F \). This means getting \( F \) alone on one side of the equation.

From our previous step, we have:

\[ r^2 = \frac{M m}{F} \]

To isolate \( F \), we can multiply both sides of the equation by \( F \):

\[ F r^2 = M m \]

Now, \( F \) is no longer in the denominator, making it easier to isolate.
Solve for Variable
After isolating the variable, the next step is to solve for it entirely. We need to perform operations that leave our variable of interest all by itself.

From the equation:

\[ F r^2 = M m \]

To solve for \( F \), divide both sides by \( r^2 \):

\[ F = \frac{M m}{r^2} \]

This leaves \( F \) isolated on one side, giving us the solution in a simplified form.
Rearrange Equations
Understanding how to rearrange equations is a core skill in algebra. This skill allows you to manipulate equations to isolate variables or convert equations to more usable forms.

Initially, our formula is:

\[ r = \sqrt{\frac{M m}{F}} \]

Through a series of rearrangements, including squaring both sides and isolating variables, we simplified it to:

\[ F = \frac{M m}{r^2} \]

Rearranging equations often involves several steps: multiplying, dividing, adding, or subtracting terms to get the desired variable by itself. Practice makes perfect, so keep solving, and these steps will become second nature.

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Most popular questions from this chapter

The length of the diagonal of a box is given by $$ D=\sqrt{L^{2}+W^{2}+H^{2}} $$ where \(L, W,\) and \(H\) are, respectively, the length, width, and height of the box. Find the length of the diagonal \(D\) of a box that is \(4 \mathrm{ft}\) long, \(2 \mathrm{ft}\) wide, and \(3 \mathrm{ft}\) high. Give the exact value, and then round to the nearest tenth of a foot.

Rationalize each denominator. Assume that all variables represent positive real numbers and that no denominators are 0. $$ \frac{5}{3 \sqrt{r}+\sqrt{s}} $$

Simplify. Assume that all variables represent positive real numbers. $$ \sqrt[3]{\frac{2}{3}} $$

Find the equation of a circle satisfying the given conditions. Center: (-8,-5)\(;\) radius: \(\sqrt{5}\)

Rationalize each denominator. Assume that all variables represent positive real numbers and that no denominators are 0. $$ \frac{\sqrt{2}-\sqrt{3}}{\sqrt{6}-\sqrt{5}} $$
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