Problem 52 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 51-54: Solve for the specified variable. \(F=g \frac{m M}{d^{2}}\) for \(d\) (Newton's law of gravitation)

Step-by-Step Solution

Answer

\( d = \sqrt{\frac{g m M}{F}} \)

Step 1: Identify the Equation

The given equation is Newton's law of gravitation: \( F = g \frac{m M}{d^2} \). Our task is to solve this equation for the variable \( d \).

Step 2: Rearrange Equation for \( d^2 \)

To isolate \( d^2 \), multiply both sides of the equation by \( d^2 \) to get: \( F d^2 = g m M \).

Step 3: Divide by \( F \)

Now, divide both sides by \( F \) to further isolate \( d^2 \): \( d^2 = \frac{g m M}{F} \).

Step 4: Take Square Root

To solve for \( d \), take the square root of both sides: \( d = \sqrt{\frac{g m M}{F}} \). This expression gives us \( d \) as the subject of the equation.

Key Concepts

Newton's Law of Gravitation

Newton's Law of Gravitation is a fundamental principle that explains the attractive force between two masses. It's stated as: \[F = g \frac{m M}{d^2}\] Where: - \(F\) is the force between the masses. - \(g\) is the gravitational constant, a fixed number that relates to the strength of gravity. - \(m\) and \(M\) are the masses of the two bodies. - \(d\) is the distance between the centers of the two masses.This formula helps model real-world scenarios like the gravitational pull of the Earth or the attraction between celestial bodies. A crucial point to understand is that gravity acts over a distance, decreasing in strength as the distance \(d\) increases. This law is vital in physics and astronomy, providing insight into how objects interact through gravity in the universe.

Variable Isolation

Variable isolation is crucial in algebra, especially when rearranging equations to solve for a specific variable. This technique involves manipulating an equation so that the variable of interest stands alone on one side of the equation. In our exercise, we need to isolate \(d\) in the equation \(F = g \frac{m M}{d^2}\). Here's a simple breakdown of the steps:

• Identify the variable: Recognize the variable you need to solve for; in our case, it's \(d\).
• Rearrange the terms: Use algebraic operations to move other terms away from \(d^2\). Multiply both sides by \(d^2\) first to get \(F d^2 = g m M\).
• Further simplify: Divide each side by \(F\) to completely isolate \(d^2\), yielding \(d^2 = \frac{g m M}{F}\).

By isolating the variable, you make it possible to lead the equation towards the desired form. This step-by-step process is foundational for any problem-solving scenarios involving equations.

Square Root Method

The Square Root Method is used to solve equations where the variable is squared. After isolating \(d^2\), we use this method to solve for \(d\) in the equation \(d^2 = \frac{g m M}{F}\). Here's how it works:

• Isolate the squared variable: First, ensure that the squared variable is on one side of the equation by itself.
• Apply the square root: Take the square root of both sides of the equation to solve for the variable, resulting in \(d = \sqrt{\frac{g m M}{F}}\).
• Consider both solutions: Remember, the square root of a number has both positive and negative solutions. In physical contexts like this one, only the positive value makes sense since distance cannot be negative.

The Square Root Method is simple yet powerful, allowing you to solve quadratic expressions efficiently. It's often used in algebraic contexts where squaring and roots are present.