Problem 35 - Algebra 2

Question

Physics The formula \(F=\frac{G m, m_{2}}{r^{2}}\) relates the gravitational force \(F\) between an object of mass \(m_{1}\) and an object of mass \(m_{2}\) separated by distance \(r\) . \(G\) is a constant known as the constant of gravitation. Solve the formula for \(r\) . Rationalize the denominator.

Step-by-Step Solution

Answer

By rearranging the equation and rationalizing the denominator, the result of solving the equation for \(r\) is \(r=\frac{\sqrt{FGm_{1}m_{2}}}{F}\).

Step 1: Rearrange the Formula to Solve for \(r\)

To solve for \(r\) in the equation \(F=\frac{G m_{1} m_{2}}{r^{2}}\), first, you will multiply both sides of the equation by \(r^{2}\) to isolate \(r^{2}\) on one side of the equation, resulting in \(Fr^{2}=Gm_{1}m_{2}\). Next, you will divide both sides of the equation by \(F\) to solve for \(r^{2}\) which will produce \(r^{2}=\frac{Gm_{1}m_{2}}{F}\).

Step 2: Take the Square Root of Both Sides

With the formula \(r^{2}=\frac{Gm_{1}m_{2}}{F}\), the next step is to isolate \(r\) by taking the square root of both sides of the equation. This produces \(r=\sqrt{\frac{Gm_{1}m_{2}}{F}}\).

Step 3: Rationalize the Denominator

To rationalize the denominator in the equation \(r=\sqrt{\frac{Gm_{1}m_{2}}{F}}\), you would multiply both the numerator and the denominator inside the square root by \(F\). The resulting equation would be \(r=\sqrt{\frac{FGm_{1}m_{2}}{F^{2}}}=\frac{\sqrt{FGm_{1}m_{2}}}{F}\).

Key Concepts

Solving for Variables

In physics, solving for variables is a common task that allows us to find the unknown values in equations. In the gravitational force equation, we start with the equation \( F = \frac{G m_1 m_2}{r^2} \). Our goal is to find \( r \), the distance between two masses. To do this, we need to isolate \( r \) on one side of the equation. We begin by multiplying both sides of the equation by \( r^2 \) to move \( r^2 \) to the side where \( F \) is:

• Step 1: Multiply both sides by \( r^2 \)
• Step 2: Rearrange to get \( Fr^2 = G m_1 m_2 \)

Next, divide both sides by \( F \) to further isolate \( r^2 \):

• Step 3: Divide both sides by \( F \)
• This results in: \( r^2 = \frac{G m_1 m_2}{F} \)

This step-by-step algebraic manipulation helps us solve for the variable \( r \) in a clear and logical manner.

Rationalizing Denominators

Rationalizing denominators is a mathematical process used to eliminate radicals from the denominator of a fraction. In the expression \( r = \sqrt{\frac{G m_1 m_2}{F}} \), we have a square root in the numerator and the denominator. Our objective is to make the denominator a rational number.To rationalize the denominator, we multiply both the numerator and the denominator by \( F \), thus eliminating the radical in the denominator. This is important because rationalizing helps simplify the expression and makes it easier to understand or compute:

• Multiply both the numerator and the denominator by \( F \)
• Transform the expression into: \( r = \frac{\sqrt{F G m_1 m_2}}{F} \)

This process ensures that the final expression is simpler and more suitable for further calculations.

Gravitational Constant

The gravitational constant, denoted as \( G \), is a key part of the gravitational force equation. The constant \( G \) is approximately \(6.674 \times 10^{-11} \, \text{Nm}^2\, \text{kg}^{-2}\) and defines the strength of gravity. It reflects how much two masses will attract each other in the universe due to gravity. This constant is fundamental in physics, especially in calculations involving gravitational forces:

• \( G \) is a crucial factor in the equation \( F = \frac{G m_1 m_2}{r^2} \)
• Plays a key role in determining the magnitude of the gravitational force

Understanding \( G \) allows us to calculate the gravitational force between two objects, which is at the core of many physical phenomena.

Algebraic Manipulation

Algebraic manipulation involves using various algebraic techniques to reorganize and solve mathematical equations. This is a critical skill in working with formulas like the gravitational force equation.When manipulating the equation \( F = \frac{G m_1 m_2}{r^2} \), we need to move terms and factors appropriately to solve for a variable. Here’s how we do it:

• Step 1: Rearrange the equation to address the desired variable (such as \( r \))
• Step 2: Simplify expressions by multiplying or dividing both sides to eliminate unwanted terms
• Step 3: Use mathematical operations, like taking square roots, to further simplify

Algebraic manipulation transforms complex equations into more manageable forms, enabling us to understand and solve problems effectively.