Problem 7 - Elementary Algebra

Question

Translate the following sentences into a mathematical formula. Every particle of matter in the universe attracts every other particle with a force, \(F\), that is directly proportional to the product of the masses, \(m_{1}\) and \(m_{2}\), of the particles and inversely proportional to the square of the distance, \(d\), between them.

Step-by-Step Solution

Answer

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

Step 1: Identify the Relationship

The problem states that the force is directly proportional to the product of the masses and inversely proportional to the square of the distance. In mathematical terms, this means that as one quantity increases, the force increases (direct proportionality), and as the other quantity increases, the force decreases (inverse proportionality).

Step 2: Express Direct Proportionality

Express the direct proportionality as: \[ F ext{ is proportional to } m_1 imes m_2 \]This means that the force increases with an increase in the product of masses.

Step 3: Express Inverse Proportionality

Express the inverse proportionality as: \[ F ext{ is proportional to } \frac{1}{d^2} \]This indicates that the force decreases as the square of the distance increases.

Step 4: Combine Both Proportionalities

Combine the expressions for direct and inverse proportionality into a single equation:\[ F ext{ is proportional to } \frac{m_1 imes m_2}{d^2} \]

Step 5: Introduce the Proportionality Constant

Replace the proportion with an equality by introducing a constant of proportionality, known as the gravitational constant, denoted by \(G\):\[ F = G \frac{m_1 imes m_2}{d^2} \] This equation represents Newton's law of universal gravitation.

Key Concepts

Direct Proportionality

In the context of Newton's Law of Universal Gravitation, **direct proportionality** refers to how the force of gravity between two objects is related to their masses.
Simply put, if you increase the mass of either or both objects, the gravitational force between them also increases.
It's like having stronger magnets; the more massive the objects, the stronger the pull between them.

• Mathematically, we express this as: \[F \text{ is proportional to } m_1 \times m_2\]
• This equation translates to: "The force \(F\) is directly proportional to the product of masses \(m_1\) and \(m_2\)".
• It implies that doubling the mass of one object while keeping the other constant doubles the gravitational force.

Direct proportionality is a crucial concept because it helps us understand how changes in mass affect gravitational attraction.

Inverse Proportionality

**Inverse proportionality** is the concept that describes how gravitational force decreases as the distance between two objects increases.
In the case of Newton's Law, the force is inversely proportional to the square of the distance between the masses. This means that the further apart two objects are, the weaker the gravitational pull between them.

• This relationship is expressed mathematically as: \[F \text{ is proportional to } \frac{1}{d^2}\]
• Essentially, if you double the distance \(d\), the force \(F\) is reduced to one-fourth.
• This square relationship suggests a rapid decrease in force with distance.

Inverse proportionality helps us comprehend why planets stay in orbit rather than crashing into the sun or flying into space.
It illustrates the balance of space distances and gravitational pull.

Gravitational Constant

The **gravitational constant**, often denoted as \(G\), is a key player in the formula for calculating gravitational force.
It represents the proportionality constant that bridges the equation of gravity, turning the proportionality into an equality.
Here’s what you should know about it:

• The **gravitational constant** ensures that the units on both sides of the equation are consistent.
• Newton's law can be represented as:\[F = G \frac{m_1 \times m_2}{d^2}\]where \(F\) is the gravitational force, \(m_1\) and \(m_2\) are the masses, and \(d\) is the distance between the centers of the two masses.
• This constant \(G\) has a fixed value of approximately \(6.674 \times 10^{-11} \, \text{Nm}^2/\text{kg}^2\).

The gravitational constant provides a way to quantify gravitational force in scientific terms, making it foundational for studies in physics and astronomy.