Question

For a point on the surface of the Moon determine the ratio of the acceleration of gravity due to the mass of the Earth to the acceleration of gravity due to the mass of the Moon.

Short answer

The ratio of Earth's gravitational acceleration on the Moon to Moon's own gravitational acceleration is approximately 0.00167.

Step-by-step solution

Understand the Gravitational Force Equation

To determine the gravitational acceleration due to a mass, we use Newton's Law of Universal Gravitation: \[ F = \frac{G \, m_1 \, m_2}{r^2} \]where \( F \) is the force between the masses, \( G \) is the gravitational constant, \( m_1 \) and \( m_2 \) are the masses, and \( r \) is the distance between the centers of the two masses. The gravitational acceleration \( a \) is computed by dividing the gravitational force by the other mass.

Calculate the Acceleration Due to Earth's Gravity

For the acceleration due to Earth's gravity at the Moon: \[ a_{earth} = \frac{G \, M_{earth}}{d^2} \]where \( M_{earth} \) is the mass of the Earth and \( d \) is the distance from the Moon to Earth.

Calculate the Acceleration Due to Moon's Gravity

The acceleration due to Moon's gravity at its surface:\[ a_{moon} = \frac{G \, M_{moon}}{R_{moon}^2} \]where \( M_{moon} \) is the mass of the Moon and \( R_{moon} \) is the radius of the Moon.

Set Up the Ratio

To find the ratio of the acceleration due to Earth's gravity to that due to Moon's gravity, set up the ratio:\[ \text{Ratio} = \frac{a_{earth}}{a_{moon}} = \frac{\frac{G \, M_{earth}}{d^2}}{\frac{G \, M_{moon}}{R_{moon}^2}} \]

Simplify the Ratio

Cancel out \( G \) from the numerator and the denominator:\[ \text{Ratio} = \frac{M_{earth} \times R_{moon}^2}{M_{moon} \times d^2} \]

Plug in Values

Use known values: - \( M_{earth} \approx 5.97 \times 10^{24} \, kg \)- \( M_{moon} \approx 7.34 \times 10^{22} \, kg \)- \( R_{moon} \approx 1.74 \times 10^6 \, m \)- \( d \approx 3.84 \times 10^8 \, m \)Plug these into the equation:\[ \text{Ratio} = \frac{(5.97 \times 10^{24} \, kg) \times (1.74 \times 10^6 \, m)^2}{(7.34 \times 10^{22} \, kg) \times (3.84 \times 10^8 \, m)^2} \]

Calculate the Result

Calculate the ratio:\[ \text{Ratio} \approx \frac{5.97 \times 10^{24} \times 3.0276 \times 10^{12}}{7.34 \times 10^{22} \times 1.4736 \times 10^{17}} \]\[ \text{Ratio} \approx \frac{18.08 \times 10^{36}}{10.82 \times 10^{39}} \approx 0.00167 \]

Key concepts

Newton's Law of Universal Gravitation

Newton's Law of Universal Gravitation is a fundamental principle that describes how objects attract each other with a force that is proportional to the product of their masses and inversely proportional to the square of the distance between their centers. The mathematical expression of this law is given by:\[ F = \frac{G \, m_1 \, m_2}{r^2} \]Where:

• \( F \) is the gravitational force between two objects.
• \( G \) is the gravitational constant \((6.674 \times 10^{-11} \text{N} \cdot \text{m}^2/\text{kg}^2)\).
• \( m_1 \) and \( m_2 \) are the masses of the two objects.
• \( r \) is the distance between the centers of these two masses.

This law explains why objects are drawn towards each other, such as the Earth pulling objects towards it, or how the Moon's gravity affects the tides on Earth. The gravitational acceleration an object feels due to another object depends on its mass and the distance between them.

Mass of Earth and Moon

To understand gravitational interactions between celestial bodies, it's crucial to know their masses. The Earth has a significantly larger mass than the Moon. Specifically:

• Mass of Earth (\( M_{earth} \)): approximately \(5.97 \times 10^{24} \text{ kg}\).
• Mass of Moon (\( M_{moon} \)): approximately \(7.34 \times 10^{22} \text{ kg}\).

These mass values are critical in calculating gravitational forces and subsequent accelerations. Because the Earth's mass is about 81 times that of the Moon, its gravitational pull is significantly stronger. This difference in mass is one of the reasons why Earth's gravitational effects extend further into space compared to the relatively smaller gravitational influence of the Moon.

Gravity on the Moon

Gravity on the Moon is quite different from gravity on Earth due to its smaller size and mass. The formula used to calculate the acceleration due to the Moon’s gravity is:\[ a_{moon} = \frac{G \, M_{moon}}{R_{moon}^2} \]Where:

• \( G \) is the gravitational constant.
• \( M_{moon} \) is the Moon’s mass.
• \( R_{moon} \) is the radius of the Moon, approximately \(1.74 \times 10^6 \text{ m}\).

On the surface of the Moon, this results in a gravitational acceleration of roughly \(1.62 \text{ m/s}^2\), which is much less than the \(9.81 \text{ m/s}^2\) experienced on Earth. This is why astronauts hop in a floating manner when moving across the Moon's surface.

Distance from Earth to Moon

The average distance from the Earth to the Moon is crucial in calculating the gravitational influence between them. This distance is approximately \(3.84 \times 10^8 \text{ m}\).Such a large separation means that while Earth's gravity does affect the Moon, it is less intense than if they were closer. The significant distance also contributes to why the Moon doesn’t crash into Earth or drift away, maintaining a stable orbit due to the equilibrium of gravitational forces.Understanding these distances helps explain phenomena such as lunar phases, eclipses, and even tidal effects on Earth, all influenced by the gravitational relationship between our planet and its natural satellite.