Problem 45 - Algebra 2

Question

According to the Law of Universal Gravitation, the attractive force \(F\) in Newtons between any two bodies in the universe is directly proportional to the product of the masses \(m_{1}\) and \(m_{2}\) in kilograms of the two bodies and the product of the masses \(m_{1}\) and \(m_{2}\) in kilograms of the two bodies and inversely proportional to the square of the distance \(d\) in meters between the bodies. That is, \(F=G \frac{m_{1} m_{2}}{d_{2}} . G\) is the universal gravitational constant. Its value is \(6.67 \times 10^{-11} \mathrm{Nm}^{2} / \mathrm{kg}^{2}\). The distance between Earth and the Moon is about \(3.84 \times 10^{8}\) meters. The mass of the Moon is \(7.36 \times 10^{22}\) kilograms. The mass of Earth is \(5.97 \times 10^{24}\) kilograms. What is the gravitational force that the Moon and Earth exert upon each other?

Step-by-Step Solution

Answer

The gravitational force between Earth and Moon is approximately \(1.99 \times 10^{20}\) Newtons.

Step 1: Identifying Known Values

We are given the distance between the Earth and the Moon, which is approximately \( d = 3.84 \times 10^8 \) meters. The mass of the Earth is \( m_1 = 5.97 \times 10^{24} \) kg, and the mass of the Moon is \( m_2 = 7.36 \times 10^{22} \) kg. The universal gravitational constant \( G \) is \( 6.67 \times 10^{-11} \mathrm{Nm}^2 / \mathrm{kg}^2 \).

Step 2: Writing the Formula

The formula for the gravitational force between two bodies is: \[ F = G \frac{m_1 m_2}{d^2} \]

Step 3: Substituting Known Values into the Formula

Now we substitute the known values into the formula: \[ F = (6.67 \times 10^{-11}) \frac{(5.97 \times 10^{24}) (7.36 \times 10^{22})}{(3.84 \times 10^8)^2} \]

Step 4: Calculating the Gravitational Force

Performing the calculations:1. Calculate the product of the masses: \( (5.97 \times 10^{24}) \times (7.36 \times 10^{22}) = 4.39512 \times 10^{47} \).2. Calculate the square of the distance: \( (3.84 \times 10^8)^2 = 1.47456 \times 10^{17} \).3. Divide the product of the masses by the square of the distance: \( \frac{4.39512 \times 10^{47}}{1.47456 \times 10^{17}} = 2.9804 \times 10^{30} \).4. Finally, multiply by \( G \): \( 6.67 \times 10^{-11} \times 2.9804 \times 10^{30} = 1.989 \times 10^{20} \). Thus, the gravitational force \( F \approx 1.99 \times 10^{20} \) Newtons.

Key Concepts

Gravitational Force

Gravitational force is the invisible glue that holds the universe together. It is an attractive force that acts between any two objects that have mass. The formula to calculate this force is given by Newton's universal law of gravitation:

• \( F = G \frac{m_1 m_2}{d^2} \)

Here, \( F \) is the gravitational force in Newtons, \( G \) is the gravitational constant, \( m_1 \) and \( m_2 \) are the masses of the two objects, and \( d \) is the distance between the centers of the two objects.
This force is always present, pulling objects toward each other. It affects everything from the falling of a stone to the orbits of planets. This concept is crucial for understanding how celestial bodies like the Earth and the Moon interact.
The force becomes stronger with increasing mass, meaning more massive objects exert a greater attraction. Conversely, it gets weaker with increased distance, which is why distant celestial bodies have less gravitational influence on each other.

Mass

Mass is a fundamental property of matter, indicating how much matter is contained in an object. It is usually measured in kilograms. In the universal gravitation equation, mass plays a significant role. The Earth and Moon's masses are crucial in computing the gravitational pull they exert on one another.
Mass is distinct from weight, although they are often confused. Weight is the gravitational force acting on an object's mass. Hence, an object's mass is constant, but its weight can change depending on the gravitational pull it experiences.
Objects with larger mass exert and experience stronger gravitational forces. This is why Earth, being much larger than the Moon in mass, has a more significant gravitational pull. Knowing the right mass values is essential in accurate gravitational force calculations.

Distance

Distance between two objects plays a pivotal role in determining the gravitational force. The formula shows that gravitational force is inversely proportional to the square of the distance between the objects \( \left( F \propto \frac{1}{d^2} \right) \). Thus, if the distance is doubled, the gravitational force becomes one-fourth as strong.
In our example with Earth and the Moon, the center-to-center distance is about \( 3.84 \times 10^8 \) meters. This large distance explains why the gravitational force, while enormous, does not pull the Moon crashing into Earth. The square of the distance \( d^2 \) significantly reduces the gravitational pull between objects over vast distances, enabling stable orbits in space.
Accurate distance measurement is crucial for precise calculations in gravitational physics. Even slight changes in distance can significantly alter the gravitational force.

Gravitational Constant

The gravitational constant \( G \) is a crucial part of the universal gravitation formula. It connects the other variables to give a measure of the gravitational force between two masses. Its value is a constant \( 6.67 \times 10^{-11} \) Newton meter squared per kilogram squared, reflecting how weak gravitational effects are on smaller scales.
This constant was first introduced by Sir Isaac Newton and later experimentally determined by Henry Cavendish. It remains vital for calculations beyond Earth and helps physicists comprehend forces within our universe.
In gravitational equations, \( G \) ensures the measurements fit with the units of force, aligning mass and distance to give a result in Newtons. The constancy of \( G \) implies that the law of gravitation holds universally, not just on Earth but across the cosmos.