Question

A typical adult human has a mass of about 70 kg. (a) What force does a full moon exert on such a human when it is directly overhead with its center 378,000 km away? (b) Compare this force with the force exerted on the human by the earth.

Short answer

(a) The moon exerts about 1.29 x 10^-4 N; (b) Earth exerts a much greater force at 686.7 N.

Step-by-step solution

Understand Gravitational Force Formula

The force exerted by a celestial body like the moon on an object is given by Newton's law of universal gravitation: \[ F = \frac{G \times m_1 \times m_2}{r^2} \]where \( F \) is the gravitational force, \( G \) is the gravitational constant \((6.674 \times 10^{-11} \, \text{Nm}^2/\text{kg}^2)\), \( m_1 \) is the mass of the moon, \( m_2 \) is the mass of the human, and \( r \) is the distance between the centers of the two masses.

Input Known Values for Moon-Human Interaction

For this calculation, use:- \( m_1 = 7.342 \times 10^{22} \, \text{kg} \) (mass of the moon),- \( m_2 = 70 \, \text{kg} \) (mass of the human),- \( r = 378,000 \, \text{km} = 3.78 \times 10^8 \, \text{m} \).Insert these values into the gravitation equation to find \( F \) exerted by the moon.

Calculate the Force Exerted by the Moon

Using the formula:\[ F = \frac{6.674 \times 10^{-11} \, \text{Nm}^2/\text{kg}^2 \times 7.342 \times 10^{22} \, \text{kg} \times 70 \, \text{kg}}{(3.78 \times 10^8 \, \text{m})^2} \]\[ F = 1.29 \times 10^{-4} \, \text{N} \].Thus, the force exerted by the full moon on the human is approximately \( 1.29 \times 10^{-4} \, \text{N} \).

Calculate the Force Exerted by the Earth

Now, use Earth's gravitational force using:\[ F = m_2 \times g \]where \( g = 9.81 \, \text{m/s}^2 \) is Earth's gravitational acceleration.\[ F = 70 \, \text{kg} \times 9.81 \, \text{m/s}^2 = 686.7 \, \text{N} \].The gravitational force exerted by the Earth on the human is approximately \( 686.7 \, \text{N} \).

Compare the Two Forces

Compare the gravitational force exerted by the moon with that by the Earth.The force from the moon \((1.29 \times 10^{-4} \, \text{N})\) is much smaller than the force from the Earth \((686.7 \, \text{N})\).\[ \frac{F_{\text{moon}}}{F_{\text{earth}}} = \frac{1.29 \times 10^{-4}}{686.7} \approx 1.88 \times 10^{-7} \].The force due to the moon is negligible compared to the force due to Earth.

Key concepts

Newton's Law of Universal Gravitation

Understanding the basics of gravitational attraction is essential in physics. Newton's law of universal gravitation describes how all objects in the universe attract each other with a force. This force is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers.

The formula is expressed as: \[ F = \frac{G \times m_1 \times m_2}{r^2} \]where:

• \( F \) is the gravitational force between the objects
• \( G \) is the gravitational constant, \( 6.674 \times 10^{-11} \, \text{Nm}^2/\text{kg}^2 \)
• \( m_1 \) is the mass of the first object
• \( m_2 \) is the mass of the second object
• \( r \) is the distance between the centers of the two objects

This law applies universally, from the attraction between celestial bodies like Earth and the moon, to the forces between objects on Earth itself.

Mass

Mass is a measure of the amount of matter in an object and it remains constant regardless of its location in the universe. It differs from weight, which is the gravitational force experienced by an object in a gravitational field.
- **Constant Property**: Mass does not change with the object's environment, unlike weight which can vary depending on gravitational pull. - **Units**: The standard unit for mass in the metric system is the kilogram (kg). - **Impact on Gravitational Force**: Greater mass means a greater gravitational force exerted, according to Newton's law. More massive objects will exert and experience stronger gravitational attractions.
In the universe, mass serves as one of the key quantities determining the gravitational interaction between objects. For example, the mass of a human (such as 70 kg) is critical in calculating the gravitational force exerted by larger celestial bodies like the moon.

Gravitational Constant

The gravitational constant \( G \) is a fundamental constant that plays a crucial role in calculating gravitational force between two masses. It helps establish the relationship between mass, force, and distance in Newton's law of universal gravitation.
- **Value**: \( G = 6.674 \times 10^{-11} \, \text{Nm}^2/\text{kg}^2 \) is considered one of the smallest constants in physics. It indicates how weak the gravitational force is compared to other forces like electromagnetism.- **Importance**: Without \( G \), we wouldn't be able to quantify the gravitational pull between objects the way we do.- **Historical Aspect**: Henry Cavendish first measured \( G \) in the 18th century through his famous torsion balance experiment, thereby allowing for the calculation of Earth's mass.
The gravitational constant is pivotal for understanding the forces that keep planets in orbit and govern the dynamics of galaxies, stars, and other celestial phenomena.

Moon-Earth Comparison

When we compare the gravitational force exerted by the moon and Earth on a human, we see a striking difference. The force exerted on a person standing on the Earth's surface is vastly greater than the force exerted by the moon.
- **Moon's Force**: Even when the moon is directly overhead, its gravitational pull on a human is relatively minuscule. For a 70 kg person, it's around \( 1.29 \times 10^{-4} \, \text{N} \).- **Earth's Force**: The gravitational force earth exerts on the same person is approximately \( 686.7 \, \text{N} \). This is the force we feel as weight due to Earth's gravitational acceleration, \( g = 9.81 \, \text{m/s}^2 \).- **Ratio**: The comparison shows that the moon's gravitational influence on a human is almost negligible. The ratio, \( \frac{F_{\text{moon}}}{F_{\text{earth}}} \approx 1.88 \times 10^{-7} \), illustrates just how dominant Earth's gravitational pull is.
Such comparisons highlight the overwhelming influence Earth's mass has on gravitational interactions experienced by objects on its surface.