Question

Challenge Problem Gravitational Force According to Newton's Law of Universal Gravitation, the attractive force \(F\) between two bodies is given by $$F=G \frac{m_{1} m_{2}}{r^{2}}$$ where \(m_{1}, m_{2}=\) the masses of the two bodies \(r=\) distance between the two bodies \(G=\) gravitational constant \(=6.6742 \times 10^{-11}\) newtons " meter \(^{2}\). kilogram \(^{-2}\) Suppose an object is traveling directly from Earth to the moon. The mass of Earth is \(5.9742 \times 10^{24}\) kilograms, the mass of the moon is \(7.349 \times 10^{22}\) kilograms, and the mean distance from Earth to the moon is 384,400 kilometers. For an object between Earth and the moon, how far from Earth is the force on the object due to the moon greater than the force on the object due to Earth?

Short answer

The object is approximately 345,800 kilometers from Earth when the moon's gravitational force is stronger.

Step-by-step solution

Understand The Gravitational Force Equation

According to Newton's Law of Universal Gravitation, the force of gravity between two masses, m_{1} and m_{2}, separated by a distance r is given by: \[ F = G \frac{m_{1} m_{2}}{r^{2}} \] where G is the gravitational constant \( 6.6742 \times 10^{-11} \) N m² kg⁻².

Set Up The Forces

Let the distance from Earth to the object be x kilometers. Then, the distance from the moon to the object will be (384,400 - x) kilometers. Express the forces acting on the object due to Earth (F_E) and the moon (F_M).

Express Earth's and Moon's Forces

The force exerted by the Earth on the object is given by: \[ F_E = G \frac{m_{\text{Earth}} m_{\text{object}}}{x^2} \] The force exerted by the moon on the object is given by: \[ F_M = G \frac{m_{\text{moon}} m_{\text{object}}}{(384,400 - x)^2} \]

Compare The Forces

Set the forces equal to one another to find the point where the gravitational pull of the moon on the object is stronger than that of Earth: \[ G \frac{m_{\text{Earth}} m_{\text{object}}}{x^2} = G \frac{m_{\text{moon}} m_{\text{object}}}{(384,400 - x)^2} \] Simplifying the equation (G and m_{\text{object}} cancel out): \[ \frac{m_{\text{Earth}}}{x^2} = \frac{m_{\text{moon}}}{(384,400 - x)^2} \]

Substitute The Masses

Substitute the known values for the masses of Earth and the moon: \[ \frac{5.9742 \times 10^{24}}{x^2} = \frac{7.349 \times 10^{22}}{(384,400 - x)^2} \]

Solve For x

Cross-multiplying to remove the fractions: \[ 5.9742 \times 10^{24} (384,400 - x)^2 = 7.349 \times 10^{22} x^2 \] Divide both sides by \( 7.349 \times 10^{22} \): \[ 81348.3 (384,400 - x)^2 = x^2 \] Taking the square root of both sides and solving for x will give two equations. Solve these equations to find the distance where the moon's gravitational force is greater than that of the Earth's.

Approximate The Result

After simplifying and solving the equation, find the distance x from Earth where the moon's gravitational pull is stronger. This distance will be approximately 345,800 kilometers from Earth.

Key concepts

Gravitational Force

Newton’s Law of Universal Gravitation explains the gravitational force between two masses. This force is an attractive force that pulls the two masses toward one another. The formula is given by: \[ F = G \frac{m_{1} m_{2}}{r^{2}} \]where:

• \( m_{1} \) and \( m_{2} \) are the masses of the two objects.
• \( r \) is the distance between the centers of the two masses.
• \( G \) is the gravitational constant.

This force is what keeps planets in orbit and objects grounded on Earth. The gravitational force increases with greater mass but decreases with greater distance. Understanding this concept is crucial for grasping the dynamics between any two objects with mass.

Gravitational Constant

The gravitational constant (\( G \)) is a key part of Newton's formula and it is a fixed value: \( 6.6742 \times 10^{-11} \text{N m}^2 \text{ kg}^{-2} \).This constant essentially scales the gravitational force to fit with the masses and distances we observe in the universe.Without it, calculating gravitational forces accurately would be impossible.The small value of \( G \) indicates that gravitational forces, compared to other forces in nature like electromagnetic forces, are relatively weak.For everyday situations, this small number ensures that the forces aren’t overpowered.

Mass and Distance Relationship

The relationship between mass, distance, and gravitational force can be summarized as follows:

• When the masses of either object increase, the gravitational force between them also increases.
• When the distance between the two objects increases, the gravitational force decreases.

Specifically, the force is inversely proportional to the square of the distance. This means that if you double the distance between two objects, the gravitational force becomes four times weaker. This inverse-square relationship is why gravitational forces between distant celestial bodies (like Earth and distant stars) are negligible, while being very significant for closer objects.

Earth and Moon Mass

The masses of Earth and the moon play a crucial role in the gravitational forces they exert. The mass of Earth is approximately \( 5.9742 \times 10^{24} \) kilograms and the moon’s mass is about \( 7.349 \times 10^{22} \) kilograms.
These large masses explain why both celestial bodies have significant gravitational pulls. The Earth’s gravitational pull is stronger due to its larger mass.However, the distance between the Earth and the moon (384,400 kilometers) needs to be considered.
To find the point where the moon's gravitational force becomes greater than the Earth's on a traveling object, we set up an equality condition and solve for the distance. Solving such problems helps us understand how varying the distances affects gravitational interactions significantly.