Problem 60 - Precalculus: Functions and Graphs

Question

The formula occurs in the indicated application. Solve for the specified variable. \(F=g \frac{m M}{d^{2}}\) for \(d \quad\) (Newton's law of gravitation)

Step-by-Step Solution

Answer

\(d = \sqrt{\frac{gmM}{F}}\)

Step 1: Identify the formula and the target variable

We have the formula for Newton's law of gravitation: \( F = g \frac{mM}{d^{2}} \), where \( F \) is the force between two masses, \( g \) is the gravitational constant, \( m \) and \( M \) are the masses, and \( d \) is the distance we need to solve for.

Step 2: Rearrange the formula

Since we need to solve for \( d \), we start by isolating \( d^2 \) on one side of the equation. Multiply both sides of the equation by \( d^2 \): \[ Fd^2 = gmM \] Now, divide both sides by \( F \) to solve for \( d^2 \): \[ d^2 = \frac{gmM}{F} \]

Step 3: Take square root to solve for d

To solve for \( d \), take the square root of both sides of the equation: \[ d = \sqrt{\frac{gmM}{F}} \] This expression gives the distance between the two masses in terms of the other variables.

Key Concepts

Solving Equations

When solving equations, the goal is to find the value of the unknown variable that satisfies the equation. In the context of Newton's law of gravitation, we are given the formula: \( F = g \frac{mM}{d^2} \). Here, we seek to solve for \( d \), which represents the distance between two masses.To solve such equations, carefully analyze what each term represents:

• \( F \) is the gravitational force.
• \( g \) is the gravitational constant.
• \( m \) and \( M \) are the masses involved.
• \( d \) is the distance we aim to find.

Understanding your variables makes solving the equation straightforward. Ensure that all calculations respect the equation structure and operations.

Rearranging Formulas

Rearranging formulas involves altering the equation to isolate the desired variable by changing its structure through basic algebraic operations. Our target is \(d\), so we'll manipulate the formula to express \(d\) explicitly.Starting from the equation:\[ F = g \frac{mM}{d^2} \]we must first clear \(d^2\) from the denominator. Multiply both sides by \(d^2\):\[ Fd^2 = gmM \]This step moves \(d^2\) out of the fraction, setting the stage for isolating \(d\).Next, isolate \(d^2\) by dividing both sides by \(F\):\[ d^2 = \frac{gmM}{F} \]This step ensures \(d^2\) stands alone on one side, making the problem simpler to tackle and leading right into the shell for the distance formula.

Distance Calculation

Once you've rearranged the formula to express \(d^2\) as \( \frac{gmM}{F} \), the final step is to solve for \(d\) by calculating the square root. This operation finds the physical distance between the two masses based on the given values.Take the square root of both sides:\[ d = \sqrt{\frac{gmM}{F}} \]This calculation effectively translates the algebraic expression into the real-world distance. Each step in finding \(d\) ensures that the manipulation of mathematical symbols represents the physical concept of distance as expressed in Newton's law of gravitation.Understanding these calculations helps grasp how changes in the variables \(g\), \(m\), \(M\), and \(F\) affect the distance \(d\). Use this expression to make informed predictions or analyses related to gravitational interactions.