Problem 68 - Calculus with Concepts in Calculus

Question

The magnitude of the gravitational force exerted on a unit mass at \((x, y, z)\) by a point mass located at the origin is given by $$ F(x, y, z)=\frac{c}{x^{2}+y^{2}+z^{2}} $$ where \(c\) is a positive constant. Describe the level surfaces of \(F\)

Step-by-Step Solution

Answer

The level surfaces are concentric spheres centered at the origin.

Step 1: Understand the Problem

We need to describe the level surfaces of the gravitational force function given by \( F(x, y, z) = \frac{c}{x^2 + y^2 + z^2} \).Level surfaces of a function are the sets of points where the function takes a constant value. In this case, we are considering surfaces where \( F(x, y, z) = k \), where \( k \) is a constant value of the force.

Step 2: Set the Function Equal to a Constant

To find the level surfaces, set \( F(x, y, z) = k \).This gives us the equation:\[ \frac{c}{x^2 + y^2 + z^2} = k \]where \( k \) is a specific constant value of the gravitational force.

Step 3: Solve for the Expression

Rearrange the equation to solve for \( x^2 + y^2 + z^2 \):\[ c = k(x^2 + y^2 + z^2) \]Now solve for \( x^2 + y^2 + z^2 \):\[ x^2 + y^2 + z^2 = \frac{c}{k} \]

Step 4: Interpretation as Level Surfaces

The equation \( x^2 + y^2 + z^2 = \frac{c}{k} \) describes a sphere centered at the origin with radius \( \sqrt{\frac{c}{k}} \). Therefore, the level surfaces of \( F(x, y, z) \) are concentric spheres centered at the origin, each corresponding to a different value of the constant \( k \).

Key Concepts

level surfaces

In mathematics, level surfaces are particularly fascinating as they help visualize a function of three variables. Think of a level surface as a slice of space.
For a given function \( F(x, y, z) \), the level surfaces represent all the combinations of \(x\), \(y\), and \(z\) where the function remains constant.
In our case, the gravitational force \( F(x, y, z) = \frac{c}{x^2 + y^2 + z^2} \) has level surfaces described by the equation \( x^2 + y^2 + z^2 = \frac{c}{k} \).
These level surfaces depict how the value of the force remains the same around the point mass in space, forming a perfect geometric shape.

mathematical functions

Mathematical functions are like special machines that return a single output for each input. These functions can describe numerous natural phenomena, like gravitational force.
Every function takes some inputs—in our instance, coordinates \((x, y, z)\)—and transforms them using a rule. This rule involves mathematical operations, such as division, in this exercise.
The function for gravitational force, \( F(x, y, z) = \frac{c}{x^2 + y^2 + z^2} \), describes how the force varies with each location around the mass.
These functions are extremely useful for predicting outcomes and mapping the behaviors of different forces in physics.

concentric spheres

Concentric spheres are circles that share the same center but have different radii. In three-dimensional space, spheres exhibit this concentric nature, making them central in understanding gravitational fields.
From our exercise, the equation describing the level surfaces \( x^2 + y^2 + z^2 = \frac{c}{k} \) forms concentric spheres around the origin.
The radius of each sphere equals \( \sqrt{\frac{c}{k}} \), with \( k \) being the constant force value.

• Every value of \( k \) corresponds to a different sphere.
• All these spheres are centered at the origin, showcasing how the gravitational force diminishes farther away from the point mass.

Concentric spheres visually represent the magnitude of the force as you move through different levels of \( k \).