Question

Express algebraically the ratio of the gravitational force on the Moon due to the Earth to the gravitational force on the Moon due to the Sun. Why, since the ratio is so small, doesn't the Sun pull the Moon away from the Earth?

Short answer

Answer: The ratio of the gravitational force on the Moon due to the Earth to the gravitational force on the Moon due to the Sun is (ME * dSM^2) / (MS * dEM^2). Despite the small ratio, the Sun does not pull the Moon away from the Earth because both the Moon and the Earth are in orbit around the Sun, and the net forces act to maintain a relatively stable orbit for the Moon around the Earth.

Step-by-step solution

Recall Newton's law of universal gravitation.

Newton's law of universal gravitation states that the gravitational force between two masses (m1 and m2) is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers (r). This is represented mathematically as: F = G * (m1 * m2) / r^2 where F is the gravitational force, G is the gravitational constant (approximately 6.674 × 10^(-11) m^3 kg^(-1) s^(-2)), m1 and m2 are the two masses, and r is the distance between them.

Calculate the gravitational force between the Earth and the Moon.

To calculate the gravitational force between the Earth (mass ME) and the Moon (mass MM), we use the above equation with their respective mass values and the average distance between the Earth and the Moon (dEM): F_Earth_Moon = G * (ME * MM) / dEM^2

Calculate the gravitational force between the Sun and the Moon.

Similarly, to calculate the gravitational force between the Sun (mass MS) and the Moon (mass MM), we use the above equation with their mass values and the average distance between the Sun and the Moon (dSM): F_Sun_Moon = G * (MS * MM) / dSM^2

Find the ratio of the gravitational forces.

Now that we have expressions for the gravitational forces between the Earth and the Moon and between the Sun and the Moon, we can find the ratio of these forces: Ratio = F_Earth_Moon / F_Sun_Moon Dividing the expressions in Steps 2 and 3, we get: Ratio = [(ME * MM) / dEM^2] / [(MS * MM) / dSM^2] We can simplify this expression further by canceling the common factor (MM) and rearranging: Ratio = (ME * dSM^2) / (MS * dEM^2)

Explain why the Sun does not pull the Moon away from the Earth.

Although the gravitational force between the Sun and the Moon is indeed greater than the force between the Earth and the Moon, this does not cause the Sun to pull the Moon away from the Earth. This is because both the Moon and the Earth are in orbit around the Sun, and the Sun's gravitational force on the Earth is similar in scale to its force on the Moon. As a result, the orbital motion of the Moon around the Earth remains relatively stable, and the Sun's gravitational force does not disrupt this motion. In conclusion, the ratio of the gravitational force on the Moon due to the Earth to the gravitational force on the Moon due to the Sun is given by the expression (ME * dSM^2) / (MS * dEM^2). Despite the small ratio, the Sun does not pull the Moon away from the Earth because both the Moon and the Earth are in orbit around the Sun, and the net forces act to maintain a relatively stable orbit for the Moon around the Earth.

Key concepts

Newton's law of universal gravitation

When we look up at the night sky and marvel at the Moon's consistent orbit around the Earth, we're witnessing the invisible hand of gravity at work. Sir Isaac Newton revolutionized our understanding of gravity with his law of universal gravitation. This fundamental principle tells us that every object in the universe exerts a gravitational pull on every other object. The strength of this pull is quite predictable: it is directly proportional to the product of the masses of the two objects and inversely proportional to the square of the distance between their centers.

For example, the force (F) of attraction that the Earth exerts on the Moon, and vice versa, can be calculated using the equation:\[\begin{equation}F = G \frac{{m_1 \times m_2}}{{r^2}}\end{equation}\]where G represents the gravitational constant (6.674 \times 10^{-11} \, m^3 kg^{-1} s^{-2}), m_1 and m_2 are the masses of the Earth and the Moon, and r is the distance between the centers of these two celestial bodies. This equation unveils the beautiful simplicity and symmetry of gravity, where every mass contributes to the cosmic dance of celestial objects.

Gravitational forces calculation

The force pulling us to the ground, keeping the Moon in orbit, and the planets revolving around the Sun can all be quantified using gravitational force calculations. Tackling such calculations requires understanding Newton's law of universal gravitation, which provides a way to predict the gravitational pull between any two masses.

To determine this force, you plug in the values for each mass and the distance between them into Newton's equation. But it's not enough to just know the formula; comprehension of the units and the enormous scales involved is also critical. For instance, when calculating the force between the Earth and the Moon, the resultant number is unthinkably large, reflecting the immense masses and distances involved. Similarly, for the Sun and the Moon, one must accurately measure the astronomical distances to ensure precise results. The remarkable part is, despite these huge numbers, the formula provides accurate computations that match our observations of the Moon’s motion around the Earth.

Celestial bodies orbit stability

You might wonder, considering the vast gravitational force the Sun exerts, why the Moon doesn't simply break away from Earth's grip and fall into the lap of the Sun. The answer lies in the concept of orbit stability. Because of the immense distance between the Sun and the Moon, and the similarity in the scale of the Sun's gravitational force on both the Earth and the Moon, they orbit the Sun as a system.

Using Newton's laws, we can understand that the Moon's orbit around the Earth is stable because it is not only affected by the Earth's gravity but also by the Sun’s gravitational force. This creates a balance, allowing the Moon to maintain a stable orbit around the Earth, despite the stronger pull from the Sun. The system's stability is a delicate dance of gravitational forces, velocities, and positions, all of which obey the laws of physics and result in the relatively stable orbits that we can observe and predict with remarkable precision. Understanding these dynamics assists scientists and astronomers in predicting celestial events and the behavior of other objects in our solar system and beyond.