Problem 40 - Single Variable Calculus

Question

The magnitude of the gravitational force exerted by the earth on a particle of mass \(m\) at a distance \(r\) from the center of the earth is $$F(r)=\left\\{\begin{array}{ll} \frac{G M m r}{R^{2}} & \text { if } rR\), and interpret your result.

Step-by-Step Solution

Answer

For \(r

Step 1: Identify the equation for r < R

First, we take the equation given for the case when the distance \(r\) is less than the Earth's radius, \(R\): \(F(r) = \frac{G M m r}{R^2}\)

Step 2: Differentiate with respect to r

We differentiate the above equation with respect to \(r\): \(\frac{dF(r)}{dr} = \frac{d}{dr}\left(\frac{G M m r}{R^2}\right)\) Using the constant rule, we can pull the constants out of the differentiation: \(\frac{dF(r)}{dr} = \frac{G M m}{R^2}\frac{d}{dr}(r)\) Now, the derivative of \(r\) with respect to \(r\) is just 1: \(\frac{dF(r)}{dr} = \frac{G M m}{R^2}\)

Step 3: Interpret the result

The result means that when a particle is inside the Earth (when its distance from the center, \(r\), is less than Earth's radius, \(R\)), the change in the gravitational force with respect to the distance is a positive constant. This suggests that the gravitational force increases linearly as the particle gets closer to the center of the Earth. #b. Derivative for r > R#

Step 1: Identify the equation for r > R

Next, we take the equation given for the case when the distance \(r\) is greater than or equal to the Earth's radius, \(R\): \(F(r) = \frac{G M m}{r^2}\)

Step 2: Differentiate with respect to r

We differentiate the above equation with respect to \(r\): \(\frac{dF(r)}{dr} = \frac{d}{dr}\left(\frac{G M m}{r^2}\right)\) Using the constant rule, we can pull the constants out of the differentiation: \(\frac{dF(r)}{dr} = G M m\frac{d}{dr}\left(\frac{1}{r^2}\right)\) Now, we differentiate \(\frac{1}{r^2}\), which is the same as \(r^{-2}\), with respect to \(r\): \(\frac{d}{dr}(r^{-2}) = -2r^{-3}\) Multiplying the derivative with the constants gives: \(\frac{dF(r)}{dr} = -2G M m r^{-3}\)

Step 3: Interpret the result

The result means that when a particle is on the surface of the Earth or beyond (when its distance from the center, \(r\), is greater than or equal to Earth's radius, \(R\)), the change in gravitational force with respect to the distance is inversely proportional to the distance cubed. The negative sign indicates that the gravitational force decreases as the distance from the center of the Earth increases, and the rate of decrease gets larger as the particle gets farther from the Earth.

Key Concepts

Gravitational Force

Gravitational force is a fundamental interaction that occurs between objects with mass. Specifically, it is an attractive force that pulls masses toward one another.
For a particle like a ball or a planet, the gravitational force it experiences due to the Earth is dependent on its mass, Earth's mass, and the distance between them. The formula used to calculate this force is derived from Newton's law of universal gravitation and is given by:

• Inside the Earth: \[F(r) = \frac{G M m r}{R^2}\]
• Outside the Earth: \[F(r) = \frac{G M m}{r^2}\]

Here, \(G\) is the gravitational constant, \(M\) is the Earth's mass, \(m\) is the mass of the object, \(R\) is the Earth's radius, and \(r\) is the distance from the Earth's center.
The gravitational force is crucial for keeping planets in orbit around stars, moons in orbit around planets, and makes everyday actions like walking on the surface of Earth possible.

Differentiation

Differentiation is a mathematical process used to find the rate of change of a function with respect to a variable. In the context of gravitational forces, differentiation helps us understand how the force changes as the distance from the Earth's center changes.
For cases where the distance \(r\) is less than Earth's radius \(R\), the differentiation of the force gives a constant positive value \(\frac{G M m}{R^2}\). This implies that as a particle moves inside the Earth, closer to the center, the gravitational force increases linearly.
In contrast, when \(r\) is greater than \(R\), the differentiation shows the force is \(-2G M m r^{-3}\). This indicates the force decreases as the particle moves away from the Earth. The negative sign tells us that the force diminishes, and the cubic relation highlights that its decline speeds up as distance increases. Thus, differentiation provides a deeper insight into how spatial variations affect gravitational interactions.

Inverse Square Law

The inverse square law is a crucial principle when it comes to understanding gravitational force outside the Earth. It states that the magnitude of a physical quantity (like gravity) is inversely proportional to the square of the distance from the source of that force.
For the gravitational force, this means:

• The force between two masses decreases four times when the distance between them doubles.
• The formula \(F(r) = \frac{G M m}{r^2}\) shows how it follows the inverse square law.

The inverse square nature is a key reason why gravity, light, and many other forces seem strong when close but weaken quickly once you move far away. Understanding this law helps you predict how gravitational forces behave as objects travel away into space or approach closer realms. It's an elegant mathematical expression of how interactions weaken with distance in an expansive universe.

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Key Concepts

Gravitational Force

Gravitational force is a fundamental interaction that occurs between objects with mass. Specifically, it is an attractive force that pulls masses toward one another.
For a particle like a ball or a planet, the gravitational force it experiences due to the Earth is dependent on its mass, Earth's mass, and the distance between them. The formula used to calculate this force is derived from Newton's law of universal gravitation and is given by:

• Inside the Earth: \[F(r) = \frac{G M m r}{R^2}\]
• Outside the Earth: \[F(r) = \frac{G M m}{r^2}\]

Here, \(G\) is the gravitational constant, \(M\) is the Earth's mass, \(m\) is the mass of the object, \(R\) is the Earth's radius, and \(r\) is the distance from the Earth's center.
The gravitational force is crucial for keeping planets in orbit around stars, moons in orbit around planets, and makes everyday actions like walking on the surface of Earth possible.

Differentiation

Differentiation is a mathematical process used to find the rate of change of a function with respect to a variable. In the context of gravitational forces, differentiation helps us understand how the force changes as the distance from the Earth's center changes.
For cases where the distance \(r\) is less than Earth's radius \(R\), the differentiation of the force gives a constant positive value \(\frac{G M m}{R^2}\). This implies that as a particle moves inside the Earth, closer to the center, the gravitational force increases linearly.
In contrast, when \(r\) is greater than \(R\), the differentiation shows the force is \(-2G M m r^{-3}\). This indicates the force decreases as the particle moves away from the Earth. The negative sign tells us that the force diminishes, and the cubic relation highlights that its decline speeds up as distance increases. Thus, differentiation provides a deeper insight into how spatial variations affect gravitational interactions.

Inverse Square Law

The inverse square law is a crucial principle when it comes to understanding gravitational force outside the Earth. It states that the magnitude of a physical quantity (like gravity) is inversely proportional to the square of the distance from the source of that force.
For the gravitational force, this means:

• The force between two masses decreases four times when the distance between them doubles.
• The formula \(F(r) = \frac{G M m}{r^2}\) shows how it follows the inverse square law.

The inverse square nature is a key reason why gravity, light, and many other forces seem strong when close but weaken quickly once you move far away. Understanding this law helps you predict how gravitational forces behave as objects travel away into space or approach closer realms. It's an elegant mathematical expression of how interactions weaken with distance in an expansive universe.