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Chapter 10: Problem 75

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

Short Answer

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

Step by step solution

01

- Square Both Sides

To isolate the variable M, begin by squaring both sides of the equation to eliminate the square root. Starting with:\[ r = \sqrt{\frac{M m}{F}} \]square both sides to get:\[ r^2 = \frac{M m}{F}\]
02

- Multiply Both Sides by F

To isolate M, multiply both sides of the equation by F, resulting in:\[ r^2 F = M m\]
03

- Solve for M

Finally, divide both sides of the equation by m to isolate M:\[ M = \frac{r^2 F}{m}\]

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Key Concepts

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

Algebraic Manipulation
Algebraic manipulation involves performing operations on an equation to simplify it or solve for a specific variable. It's like solving a puzzle where you use addition, subtraction, multiplication, and division to transform the equation step by step.

In our given example, we started with the equation:

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

This involves understanding how to remove the square root by squaring both sides. This fundamental algebraic operation helps us further simplify and solve the equation. Breaking problems into smaller steps, like squaring both sides, makes them more manageable.

In practice:
  • Always perform the same operation on both sides of an equation.
  • Try to keep the equation balanced.
Formula Rearrangement
Formula rearrangement is essential when solving for a specific variable. To rearrange, we perform a series of algebraic manipulations to change the structure of the equation.

Let's break it down. First, we squared both sides to eliminate the square root:

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

Next, we multiply both sides by F:

\(r^2 F = M m\)


This step aims to move the needed variable (M) out of the fraction. Rearranging formulas involves:
  • Identifying the variable you need to solve for.
  • Applying inverse operations (like squaring) to both sides.
  • Maintaining equation balance.
Formula rearrangement is very useful in physics and engineering when you need to rework formulas to find different quantities.
Isolating Variables
Isolating variables means getting the variable you are solving for alone on one side of the equation. This often requires a final step of dividing or multiplying.

In our example, to isolate M from:
\(r^2 F = M m\), we divided both sides by m:

\(M = \frac{r^2 F}{m}\)
This isolates M, making it the sole subject of the equation.
Key principles for isolating variables:
  • Identify the variable you need to isolate.
  • Gradually undo operations around it by performing inverse operations (e.g., if the variable is multiplied, divide).
  • Simplify step by step to keep things clear.
Isolating variables is a cornerstone in solving equations, as it helps us find exact values or expressions for specific variables.

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