Problem 76 - Intermediate Algebra

Question

Solve for the specified variable. $$ F=\frac{G m_{1} m_{2}}{r^{2}} \quad \text { for } m_{1} $$

Step-by-Step Solution

Answer

\( m_1 = \frac{F r^2}{G m_2} \)

Step 1: Understand the Equation

The given equation is the formula for gravitational force: \[ F = \frac{G m_1 m_2}{r^2} \]where \( F \) is the gravitational force, \( G \) is the gravitational constant, \( m_1 \) and \( m_2 \) are the masses of the two objects, and \( r \) is the distance between the centers of the two objects. We need to solve for \( m_1 \).

Step 2: Isolate \( m_1 \) on One Side

Start by multiplying both sides of the equation by \( r^2 \) to remove the denominator:\[ F r^2 = G m_1 m_2 \].This clears the fraction by getting \( Gm_1m_2 \) on one side of the equation.

Step 3: Solve for \( m_1 \)

Next, divide both sides of the equation by \( G m_2 \) to isolate \( m_1 \):\[ m_1 = \frac{F r^2}{G m_2} \].Now, \( m_1 \) is isolated, and you have expressed it in terms of \( F, r, G, \) and \( m_2 \).

Key Concepts

Gravitational Force Formula

The formula representing gravitational force is pivotal in physics. This formula is:\[ F = \frac{G m_1 m_2}{r^2} \]Here,

• \( F \) is the gravitational force between two masses.
• \( G \) is the gravitational constant, a fixed value that keeps equations consistent.
• \( m_1 \) and \( m_2 \) are the masses of the objects involved.
• \( r \) represents the distance between the centers of these two masses.

This equation emphasizes the inverse square principle where the force is inversely proportional to the square of the distance between the objects. This means that as distance \( r \) increases, the gravitational force \( F \) decreases, and vice versa. Grasping this formula is crucial as it forms the basis for understanding gravitation in physics.

Isolating Variables

In algebra, isolating a variable means manipulating the equation until that variable is on its own on one side of it. Here's how you isolate \( m_1 \) in the gravitational force formula:First, we aim to clear any fractions. Multiplying both sides by \( r^2 \) helps remove this, giving:\[ F r^2 = G m_1 m_2 \]Now, \( G m_1 m_2 \) is isolated on one side of the equation without a denominator.Next, to further isolate \( m_1 \), divide both sides by \( G m_2 \):\[ m_1 = \frac{F r^2}{G m_2} \]This algebraic step ensures \( m_1 \) stands alone, expressed clearly in terms of the other variables. These techniques of shifting terms and dividing systematically are valuable for solving many algebraic equations.

Algebraic Manipulation

Algebraic manipulation is a critical skill in mathematics, particularly in solving equations. It involves a series of operations such as addition, subtraction, multiplication, and division to rearrange or simplify equations. In the given problem, several manipulation strategies were used:

• Multiplying both sides of the equation by \( r^2 \): This was used to eliminate the fraction in the original formula. Fractions can complicate calculations, so removing them early simplifies the process.
• Dividing by \( Gm_2 \): This was necessary to complete the isolation of the variable \( m_1 \). By dividing, we effectively balance the equation to reveal the isolated variable on one side.

These algebraic techniques are combined to make the equation simpler and solve for the desired variable. Mastering these manipulations allows you to tackle various mathematical problems with confidence.