Problem 139 - Calculus Volume 3

Question

The mass of Earth is approximately \(6 \times 10^{27} \mathrm{~g}\) and that of the Sun is 330,000 times as much. The gravitational constant is \(6.7 \times 10^{-8} \mathrm{~cm}^{3} / \mathrm{s}^{2} \cdot \mathrm{g}\). The distance of Earth from the Sun is about \(1.5 \times 10^{12} \mathrm{~cm}\). Compute, approximately, the work necessary to increase the distance of Earth from the Sun by \(1 \mathrm{~cm}\).

Step-by-Step Solution

Answer

The work necessary is approximately \( 3.52 \times 10^{19} \text{ ergs} \).

Step 1: Understand Gravitational Force Formula

The gravitational force between two masses is given by the formula \( F = \frac{{G imes m_1 imes m_2}}{{r^2}} \), where \( G \) is the gravitational constant, \( m_1 \) and \( m_2 \) are the masses, and \( r \) is the distance between the centers of the two masses.

Step 2: Substitute Known Values for Masses and Distance

Substitute the given values: The mass of the Earth is \(6 \times 10^{27}\text{ g} \) and the mass of the Sun is \(330,000 \times 6 \times 10^{27} \text{ g} \). The gravitational constant \( G = 6.7 \times 10^{-8} \mathrm{~cm}^{3}/\mathrm{s}^{2} \cdot \mathrm{g} \), and the distance \( r = 1.5 \times 10^{12} \mathrm{~cm} \).

Step 3: Calculate Gravitational Force

Compute the force using the formula:\[F = \frac{{6.7 \times 10^{-8} \times 6 \times 10^{27} \times 330,000 \times 6 \times 10^{27}}}{{(1.5 \times 10^{12})^2}}\]Calculate the result to find the force between the Earth and the Sun.

Step 4: Determine Work Done for 1 cm Increase in Distance

The work done to move the Earth from a distance \( r \) to \( r + 1 \text{ cm} \) is given by the change in potential energy \( \, W = F \times \Delta r \), where \( \Delta r = 1 \text{ cm} \). Compute \( W \) using the force calculated in the previous step.

Key Concepts

Mass of Earth

The mass of Earth is a fundamental factor in understanding gravitational force and other planetary interactions. The Earth's mass is approximately \(6 \times 10^{27}\, \text{g}\). This is an essential value because, in physics, the mass of an object dictates how much it pulls on other objects with its gravitational field.
Earth's mass affects everything from the tides in the oceans to the movement of satellites. The hefty mass results from the accumulated atoms and molecules that compose the planet, from the core to the atmosphere.

• The mass is primarily calculated using gravitational interactions, particularly with nearby objects like the moon.
• Understanding Earth's mass helps scientists and engineers predict orbital behaviors for satellites.

When studying exercises related to gravitational force, always remember the mass represents how Earth interacts in the solar system, gravitating towards other celestial bodies.

Gravitational Constant

The gravitational constant, denoted as \( G \), is a key component in calculating gravitational force between two masses. It is approximately \( 6.7 \times 10^{-8} \, \text{cm}^3/\text{s}^2 \cdot \text{g} \). This universal constant allows us to compute forces beyond our immediate environment, across the universe.

• The gravitational constant \( G \) is universal, meaning it's the same everywhere in the universe.
• Often measured in conjunction with gravitational force experiments to confirm its consistency.

\( G \) enables scientists to connect the mass of objects to the force exerted upon one another, regardless of where they are in the cosmos. This constant is pivotal in helping to understand not just Earth and its fellows in the solar system, but also interactions at much larger scales, across galaxies.

Work in Physics

In physics, "work" is a measure of energy transfer when an object is moved over a distance by an external force. It is described by the formula \( W = F \times \Delta r \), where \( F \) is the force applied and \( \Delta r \) is the distance moved in the direction of the force.
Calculating the work done to move Earth by 1 cm from the Sun involves understanding how energy is needed to alter the gravitational interaction.

• Work is positive when the force has a component in the direction of movement.
• Influences various processes, from lifting an object to the mechanics of astronomical systems.

In the context of gravitational forces between Earth and the Sun, calculating work helps illustrate the considerable energy requirements for seemingly minor adjustments in their proximity.

Distance in Physics

In physics, 'distance' refers to the space between two points. While simple to grasp, it is a crucial element for calculations involving motion, force, and energy. In gravitational equations, like the force between the Earth and the Sun, distance is squared, \( r^2 \), which shows its profound effect on the magnitude of force.

• Small changes in distance can result in significant differences in gravitational force due to the inverse-square law.
• Distance is a scalar quantity, meaning it only has magnitude and no direction—it is always positive.

In exercises that explore moving Earth further from the Sun, even a minor increase, such as 1 cm, in this distance can illustrate very real differences in gravitational pull, work needed, and energy expenditure.