Problem 59 - Calculus with Concepts in Calculus

Question

The gravitational force \(F\) that the earth exerts on a unit mass depends on whether the mass is inside or outside the earth. Let \(R\) denote the radius of the earth and \(r\) the distance of the unit mass from the center of the earth. Then \(F\) is given by $$ F=\left\\{\begin{array}{ll} \frac{G M r}{R^{3}} & \text { for } 0

Step-by-Step Solution

Answer

Sketch a linear increase from 0 to R and a hyperbolic decrease from R to 2R.

Step 1: Identify the Equations for F

The gravitational force \( F \) is described by two different equations based on the distance \( r \) from the center of the Earth. For \( 0 < r < R \), the force is \( F = \frac{G M r}{R^3} \). For \( r \geq R \), the force is \( F = \frac{G M}{r^2} \). These equations must be evaluated separately as they apply to different parts of the graph.

Step 2: Graph F for 0 < r < R

For \( 0 < r < R \), \( F \) is a linear function of \( r \) with slope \( \frac{G M}{R^3} \). It starts from zero when \( r = 0 \) and reaches \( \frac{G M}{R^2} \) when \( r = R \). Plot this part on the graph as a straight line segment starting from the origin and increasing linearly up to \( r = R \).

Step 3: Graph F for r >= R

For \( r \geq R \), the gravitational force follows the equation \( F = \frac{G M}{r^2} \). This represents a hyperbola. At \( r = R \), the value is \( \frac{G M}{R^2} \) and decreases as \( r \) increases. Plot this part of the function starting from \( r = R \) and asymptotically approaching zero as \( r \) moves towards \( 2R \) or beyond.

Step 4: Combine the Graph Sections

Now, combine the two parts of the graph. Connect the linear increase from \( 0\) to \( R \) with the decreasing hyperbolic section from \( R \) onwards. Ensure continuity at \( r = R \), where both parts of the equation yield the same value \( \frac{G M}{R^2} \).

Step 5: Finalize the Sketch

Label the graph's axes, with \( r \) on the horizontal axis ranging from \( 0 \) to \( 2R \), and \( F \) on the vertical axis. Mark key values and transitions on the graph, such as \( r = R \) where the slope changes from linear to hyperbolic. Make sure the graph is neat and clearly displays the behavior of \( F \) across the specified range.

Key Concepts

Graph of Force Function

When analyzing physical forces, crafting a graph is an excellent way to visualize how a force like gravity changes over distance. In this exercise, we are asked to graph the gravitational force, \( F \), exerted by the Earth on a unit mass with varying distances \( r \) from the Earth's center. This graph reveals two pieces: one where the mass is inside the Earth, and one where it is on or outside the Earth's surface.

• **For \( 0 < r < R \):** The graph of \( F \) is linear, forming a straight line that begins at the origin (\( F = 0 \) at \( r = 0 \)) and ascends with a slope of \( \frac{G M}{R^3} \) reaching \( \frac{G M}{R^2} \) when \( r = R \).
• **For \( r \geq R \):** The graph changes to a hyperbolic form, decreasing continually as \( r \) grows, characterized by \( F = \frac{G M}{r^2} \). This signifies that as distance increases, the gravitational force diminishes, asymptotically approaching zero.

By combining these two graph sections, we achieve a continuous graph transitioning smoothly from a linear segment to a curve, demonstrating how gravitational force intensifies linearly until reaching the Earth's surface, then weakens rapidly with greater distances.

Piecewise Functions

Piecewise functions are vital in mathematics and physics for expressing situations where a function behaves differently over specific intervals. In our task, the force function \( F \) is a piecewise function defined for different ranges of \( r \).

• The **first piece**, \( F = \frac{G M r}{R^3} \), operates for all \( r \) between 0 and \( R \). This linear equation models how the force increases uniformly as one moves from the center of the Earth towards the surface.
• The **second piece**, \( F = \frac{G M}{r^2} \), is applicable for \( r \geq R \). Here, the function models gravitational force that follows an inverse-square law, showcasing a decrease as the distance \( r \) increases beyond the Earth's surface.

Each piece of a piecewise function serves a unique aspect of the force's behavior based on the position of the object within or outside the Earth. Understanding piecewise functions ensures clarity when interpreting function graphs that split into different segments but join at certain points.

Radius and Distance in Physics

Understanding radius and distance in physics is crucial when studying forces like gravity. For our context, the terms "radius" and "distance" pertain specifically to spatial relationships and magnitudes:

• **Radius \( R \):** This is the fixed distance from the Earth's center to its surface. A constant that provides a boundary between the behaviors of the gravitational force inside versus outside the Earth.
• **Distance \( r \):** This dynamic variable represents the distance from the Earth's center to a point of interest. It determines how the gravitational force changes for the object in question. For \( r < R \), the focus is on increasing force as you move outwards through the Earth. For \( r \geq R \), it shifts to how the force diminishes with an increase in distance past the surface.

By applying these concepts, students can predict the nature of gravitational pull experienced at different points relative to the Earth's center. These fundamental physics principles aid in calculating gravitational effects accurately across various scenarios, from probing beneath the Earth's surface to venturing beyond it.