Problem 44 - Calculus: Early Transcendentals

Question

The gravitational force exerted by the planet Earth on a unit mass at a distance \( r \) from the center of the planet is \( F(r) = \left\\{ \begin{array}{ll} \frac{GMr}{R^3} & \mbox{if \) r < R \(}\\\ \frac{GM}{r^2} & \mbox{if \) r \ge R \(} \end{array} \right.\) where \( M \) is the mass of Earth, \( R \) is its radius, and \( G \) is the gravitational constant. Is \( F \) a continuous function of \( r \)?

Step-by-Step Solution

Answer

Yes, \( F \) is a continuous function of \( r \).

Step 1: Identify Continuity Condition

A function is continuous at a point if the limit from the left equals the limit from the right, and both equal the function's value at that point. We need to determine if this condition holds at the boundary point where the formula changes, specifically where \( r = R \).

Step 2: Analyze Limit from the Left (\( r < R \))

When \( r \) approaches \( R \) from the left, we use the expression \( \frac{GMr}{R^3} \). Calculate the limit:\[\lim_{{r \to R^-}} \frac{GMr}{R^3} = \frac{GMR}{R^3} = \frac{GM}{R^2}.\]

Step 3: Analyze Limit from the Right (\( r \ge R \))

As \( r \) approaches \( R \) from the right, we use the expression \( \frac{GM}{r^2} \). Calculate the limit:\[\lim_{{r \to R^+}} \frac{GM}{r^2} = \frac{GM}{R^2}.\]

Step 4: Verify Continuity at \( r = R \)

The function value at \( r = R \) according to the second case is \( F(R) = \frac{GM}{R^2} \). Both limits at \( r = R \) from Step 2 and Step 3 result in \( \frac{GM}{R^2} \). Since both one-sided limits and the function value coincide, \( F \) is continuous at \( r = R \).

Step 5: Conclusion

Since the function \( F \) satisfies the conditions for continuity at the boundary \( r = R \), and there are no discontinuities in either of its defined ranges, the function \( F \) is continuous for all \( r \).

Key Concepts

Gravitational Force

The gravitational force, a fundamental force in physics, is responsible for the attraction between masses. The formula for gravitational force typically involves the gravitational constant, mass of the objects, and the distance between them. In this particular exercise, the gravitational force is represented as a piecewise function, reflecting different expressions based on how close the object is to Earth's center. This allows us to understand gravitational pull both inside Earth's surface and further away. The two formulas show how the force behaves linearly when near Earth (\( r < R \)) and decreases with the square of the distance when far (\( r \ge R \)). This blend ensures we appreciate the dual nature of gravitational attraction depending on proximity to Earth.

Piecewise Functions

Piecewise functions are versatile mathematical tools that define different expressions over specific ranges of the domain. In the problem provided, the gravitational force depends on whether we consider distances less than or greater than Earth’s radius. Hence, the function switches behavior around a boundary point, \( r = R \).

• For \( r < R \), the force is directly proportional to \( r \).
• For \( r \ge R \), the gravitational pull decreases inversely with the square of \( r \).

This setup allows seamless modeling of physics phenomena where conditions change at a boundary. More importantly, it lays a foundation for analyzing continuity and limits, which are keys in determining function behavior across different intervals.

Limits in Calculus

Limits are a cornerstone of calculus, enabling the analysis of functions as they approach specific points or move towards infinity. For our piecewise gravitational function, finding limits as \( r \) approaches \( R \) from both directions is crucial. In this scenario:

• \( \lim_{{r \to R^-}} \frac{GMr}{R^3} = \frac{GM}{R^2} \)
• \( \lim_{{r \to R^+}} \frac{GM}{r^2} = \frac{GM}{R^2} \)

Both limits equate to the same value, demonstrating that the function smoothly arrives at the same outcome from either side of \( R \). This mechanism of taking limits helps assess the behavior of functions near boundaries and is fundamental for confirming their continuity.

Mathematical Continuity

Continuity, an essential concept in mathematics, describes functions without interruptions or jumps between intervals. For our gravitational piecewise function, continuity at \( r = R \) was validated by showing aligned one-sided limits and function values. A continuous function maintains no sharp changes in value at any point in its domain. To check this for our function, we confirm:

• The left-hand limit equals the right-hand limit.
• Both limits coincide with the function's actual value at the boundary (\( F(R) \)).

Ensuring the piecewise definition meets these criteria confirms that the gravitational pull remains seamlessly linked across all regions. This continuous connection is critical for predictable physical interpretations in varying scenarios.