Problem 71 - Calculus Early Transcendentals: Pearson New International Edition

Question

The gravitational force exerted by the earth on an object having mass \(m\) that is a distance \(r\) from the center of the earth is $$ g(r)= \begin{cases}\frac{G M m r}{R^{3}}, & \text { if } r

Step-by-Step Solution

Answer

Yes, g is a continuous function of r.

Step 1: Understand the Function

The function for gravitational force, \( g(r) \), is defined piecewise: one expression for when the object is inside the Earth (\( r < R \)), and another when it is at or outside the Earth's surface (\( r \geq R \)). This means we need to check continuity where the function definition changes, which is at \( r = R \).

Step 2: Evaluate the Function at the Boundary

First, calculate \( g(R) \) using both sections of the piecewise function. When \( r = R \), the first part becomes \( \frac{G M m R}{R^3} = \frac{G M m}{R^2} \). For the second part, substituting \( r = R \), gives \( \frac{G M m}{R^2} \). Both expressions equal \( \frac{G M m}{R^2} \), meaning \( g(R) \) is the same from both sides.

Step 3: Ensure Continuity at the Boundary

For \( g(r) \) to be continuous at \( r = R \), the left-hand limit (as \( r \) approaches \( R \) from the left, \( r < R \)) and the right-hand limit (as \( r \) approaches \( R \) from the right, \( r \geq R \)) must be equal, and also equal to the function value at \( r = R \). From Step 2, both limits and the function value equal \( \frac{G M m}{R^2} \). Thus, \( g(r) \) is continuous at \( r = R \).

Step 4: State the Conclusion

Since the function \( g(r) \) matches at the boundary \( r = R \), with limits from both sides equaling the function value at the boundary, \( g(r) \) is a continuous function of \( r \).

Key Concepts

Piecewise Functions

Piecewise functions are mathematical expressions defined by different formulas or rules over different parts of their domain. This is exactly what happens in our equation for gravitational force, \( g(r) \). This function is separated into two pieces:

• For when the object is inside the Earth (\( r < R \)): \( \frac{G M m r}{R^{3}} \)
• For when the object is at the Earth's surface or further out (\( r \geq R \)): \( \frac{G M m}{r^{2}} \)

Each piece applies over a specific interval of \( r \), the object's distance from the Earth's center. The switch between these formulations occurs at the earth's radius (\( r = R \)). To ensure understanding, think of a piecewise function as a function that changes its rule, like a traffic light changing colors based on conditions, such as time or traffic flow. In mathematics, piecewise functions allow scientists to model complex scenarios that require different rules under different conditions.

Gravitational Constant

The gravitational constant, denoted as \( G \), is a crucial part of the gravitational force equation. It represents the strength of gravity in Newton’s law of universal gravitation, describing how masses attract each other.

• Universal value: \( G \approx 6.674 \times 10^{-11} \text{ m}^3 \text{kg}^{-1} \text{s}^{-2} \)
• Interacts with the masses \(M\) and \(m\) in the force equation

\( G \) is what makes it possible to compute gravitational forces between objects, whether they are satellites circling the Earth or distant stars affecting each other. By incorporating \( G \), our piecewise equation for gravity remains consistent and universally applicable, no matter where or when measurements are taken, making it an essential element in celestial mechanics.

Continuity in Calculus

Continuity in calculus refers to a function being unbroken or uninterrupted. A function is continuous at a point if:

• The function is defined at that point.
• The limit of the function as it approaches from both the left and right is equal to the function's value at that point.

In our exercise, \( g(r) \) needs to be continuous at the boundary \( r = R \). This is critical because at \( r = R \), the rule changes between the two parts of the piecewise function.
By checking that both expressions of the function yield the same result at \( r = R \), we ensure there are no jumps or gaps. If any mismatch occurred, \( g(r) \) would not be continuous, similar to how a crack in a sidewalk disrupts continuous walking. Being continuous allows \( g(r) \) to smoothly transition without interruptions as an object's distance from Earth's core changes across \( R \).

Limits and Continuity

Limits and continuity are closely linked concepts in calculus. A limit is the value a function approaches as the input approaches a certain point. Continuity requires that these limits from every side of a point meet the function's value at that point.

• Left-hand limit: approaching the point from the left side
• Right-hand limit: approaching from the right side
• Function value at the point

For the gravitational force function \( g(r) \), ensuring the left- and right-hand limits at the point \( r = R \) are both \( \frac{G M m}{R^2} \), and equal to \( g(R) \), confirms continuity.
In a sense, limits declare the readiness of a function to adapt smoothly at a boundary, ensuring nothing is left incomplete or undefined. Understanding limits helps confirm that no sudden jumps disrupt a function, holding true whether moving through equations or walking through a smooth, uninterrupted path in everyday scenarios.