Problem 15 - Technical Mathematics with Calculus

Question

Newton's law of gravitation states that any two bodies attract each other with a force that is inversely proportional to the square of the distance between them. The force of attraction between two certain steel spheres is \(3.75 \times 10^{-5}\) dyne when the spheres are placed \(18.0 \mathrm{cm}\) apart. Find the force of attraction when they are \(52.0 \mathrm{cm}\) apart.

Step-by-Step Solution

Answer

The force of attraction when the spheres are 52.0 cm apart is approximately 4.49 x 10^{-6} dynes.

Step 1: Identify the Variables

Let the original force be denoted as F1 and the new force as F2. The original distance is r1 = 18.0 cm and the new distance is r2 = 52.0 cm. F1 is given as \(3.75 \times 10^{-5}\) dyne. The task is to find F2 using the inverse square law.

Step 2: Apply Newton's Law of Universal Gravitation

Newton's gravitational law suggests that the force between two objects is inversely proportional to the square of the distance between them. This can be expressed as: \[ F \propto \frac{1}{r^2} \] When comparing two forces at different distances, the formula can be rearranged to show: \[ \frac{F1}{F2} = \frac{r2^2}{r1^2} \]

Step 3: Calculate the New Force

Substitute the known values of F1, r1, and r2 into the equation to solve for F2: \[ \frac{3.75 \times 10^{-5} \text{ dynes}}{F2} = \frac{(52.0 \text{ cm})^2}{(18.0 \text{ cm})^2} \] Now solve for F2 by multiplying both sides by F2 and then dividing by the fraction on the right-hand side: \[ F2 = \frac{3.75 \times 10^{-5} \text{ dynes} \times (18.0 \text{ cm})^2}{(52.0 \text{ cm})^2} \]

Step 4: Compute the Final Answer

Compute the value of F2 using a calculator: \[ F2 = \frac{3.75 \times 10^{-5} \text{ dynes} \times 18.0^2}{52.0^2} \approx 3.75 \times 10^{-5} \text{ dynes} \times \frac{324}{2704} \approx 3.75 \times 10^{-5} \times 0.1198 \approx 4.49 \times 10^{-6} \text{ dynes} \]

Key Concepts

Inverse Square Law

The inverse square law is a principle that appears in several areas of physics, but it is particularly poignant in the realm of gravity. It states that a specified physical quantity or strength is inversely proportional to the square of the distance from the source of that physical quantity. In simple terms, if you double the distance between two objects, the gravitational force becomes four times weaker; triple the distance, and it becomes nine times weaker, and so on.

When applied to gravity, this law neatly explains why planets stay in orbit and why the force of attraction between objects diminishes so drastically as they move apart. The formula that expresses this relationship is:

Gravitational Force Calculation

Calculating the gravitational force between two objects using Newton's Law of Universal Gravitation requires understanding the inverse square law, which is integral to the formula. The gravitational force (F) between two objects is directly proportional to the product of their masses (m1 and m2) and inversely proportional to the square of the distance (r) between their centers: F = G m1 m2 r^2 where G is the gravitational constant. In our textbook problem, we're given the force at one distance and asked to find it at another, utilizing the idea that the force's ratio is equal to the square of the ratios of distances. The process involves identifying variables, applying the law, substituting known values, and computing the answer. The calculation reflects the powerful scalability of the inverse square law in practical applications.

For students, the challenges often lie in correctly setting up the proportion equation and in handling the units (e.g., converting centimeters to meters if necessary). It's crucial to be meticulous with calculations to obtain the precise value of the gravitational force at different distances.

This understanding not only helps with homework problems but also provides insight into how gravity shapes the cosmos, influencing everything from the orbits of moons to the paths of comets.

Proportional Relationships in Physics

Proportional relationships are at the heart of many physics equations and help us to understand how quantities relate to one another. In physics, when we say two values are directly proportional, it means that as one value increases, the other increases at the same rate. Conversely, when we say that two values are inversely proportional, as is the case with the gravitational force and the square of the distance between two objects, increasing one value leads to a decrease in the other in a specific manner described by the inverse square law.

These proportional relationships allow us to predict and calculate various physical quantities. In the context of our gravitational force problem, the strength of the force is inversely proportional to the square of the distance. Understanding this relationship is vital to solving the problem and is an example of an underlying principle that governs many interactions in physics.

Spotting these relationships can often simplify complex problems and make it clearer why certain quantities behave the way they do. Whether dealing with dynamics, thermodynamics, or electromagnetism, a solid grasp of proportional relationships is fundamental to mastering physics concepts.