Find the ratio of the gravitational force on the Moon due to the Earth to the gravitational force on the Moon due to the Sun. (The mass of the Earth is 5.97 1024 kg, the mass of the Moon is 7.36 1022 kg, the mass of the Sun is 1.99 1030 kg, and the average Earth-Moon distance is 385,000 km, and the average distance between the Earth and the Sun is 1.50 1011 km.)
1798:1
To find the ratio of the gravitational force on the Moon due to the Earth to the gravitational force on the Moon due to the Sun, we can use Newton's law of universal gravitation.
The formula for gravitational force is:
F = G * (m1 * m2) / r^2
Where:
F is the gravitational force
G is the gravitational constant (approximately 6.67 x 10^-11 Nm^2/kg^2)
m1 and m2 are the masses of the two objects
r is the distance between the centers of the two objects
Let's calculate the gravitational forces:
Gravitational force on the Moon due to the Earth:
F1 = G * (Mass of the Moon * Mass of the Earth) / (Distance between Earth and Moon)^2
Gravitational force on the Moon due to the Sun:
F2 = G * (Mass of the Moon * Mass of the Sun) / (Distance between Sun and Moon)^2
Substituting the given values:
Mass of the Earth = 5.97 x 10^24 kg
Mass of the Moon = 7.36 x 10^22 kg
Mass of the Sun = 1.99 x 10^30 kg
Distance between Earth and Moon = 385,000 km = 385,000,000 m
Distance between Sun and Moon = 1.5 x 10^11 km = 1.5 x 10^11,000,000 m
Calculating the forces:
F1 = (6.67 x 10^-11 Nm^2/kg^2) * (7.36 x 10^22 kg) * (5.97 x 10^24 kg) / (385,000,000 m)^2
F2 = (6.67 x 10^-11 Nm^2/kg^2) * (7.36 x 10^22 kg) * (1.99 x 10^30 kg) / (1.5 x 10^11,000,000 m)^2
Now, we can find the ratio:
Ratio = F1 / F2
To find the ratio of the gravitational force on the Moon due to the Earth to the gravitational force on the Moon due to the Sun, we can use the formula for gravitational force:
F = (G * m1 * m2) / r^2
Where F is the gravitational force, G is the gravitational constant (6.67 * 10^-11 Nm^2/kg^2), m1 and m2 are the masses of the two objects, and r is the distance between the centers of the two objects.
Let's calculate the gravitational force on the Moon due to the Earth first:
F_earth-moon = (G * m_earth * m_moon) / r_earth-moon^2
F_earth-moon = (6.67 * 10^-11 Nm^2/kg^2) * (5.97 * 10^24 kg) * (7.36 * 10^22 kg) / (3.85 * 10^8 m)^2
Next, let's calculate the gravitational force on the Moon due to the Sun:
F_sun-moon = (G * m_sun * m_moon) / r_sun-moon^2
F_sun-moon = (6.67 * 10^-11 Nm^2/kg^2) * (1.99 * 10^30 kg) * (7.36 * 10^22 kg) / (1.50 * 10^11 m)^2
Now, we can find the ratio by dividing the force due to the Earth by the force due to the Sun:
Ratio = F_earth-moon / F_sun-moon
Ratio = ((6.67 * 10^-11 Nm^2/kg^2) * (5.97 * 10^24 kg) * (7.36 * 10^22 kg) / (3.85 * 10^8 m)^2) / ((6.67 * 10^-11 Nm^2/kg^2) * (1.99 * 10^30 kg) * (7.36 * 10^22 kg) / (1.50 * 10^11 m)^2)
Simplifying the expression, we get:
Ratio = ((5.97 * 7.36) / (3.85^2)) / ((1.99 * 7.36) / (1.50^2))
Calculating this expression, we find:
Ratio ≈ 0.4637
Therefore, 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 approximately 0.4637.