The drawing (not to scale) shows one alignment of the sun, earth, and moon. The gravitational force that the sun exerts on the moon is perpendicular to the force that the earth exerts on the moon. The masses are: mass of sun=1.99 × 1030 kg, mass of earth=5.98 × 1024 kg, mass of moon=7.35 × 1022 kg. The distances shown in the drawing are rSM = 1.50 × 1011 m and rEM = 3.85 × 108 m. Determine the magnitude of the net gravitational force on the moon.
To determine the magnitude of the net gravitational force on the moon, you can use the formula for gravitational force:
F = G * (m1 * m2) / r^2
where F is the gravitational force, G is the gravitational constant (approximately 6.67430 × 10^-11 N m^2 / kg^2), m1 and m2 are the masses of the two objects, and r is the distance between them.
In this case, you need to calculate the net gravitational force on the moon due to both the sun and the earth. Since the forces are perpendicular, you can add them as vectors to find the net force. The magnitude of the net force can be found using the Pythagorean theorem.
1. Calculate the force due to the sun:
F_sun = (G * m_sun * m_moon) / r_SM^2
2. Calculate the force due to the earth:
F_earth = (G * m_earth * m_moon) / r_EM^2
3. Calculate the net force on the moon:
F_net = √(F_sun^2 + F_earth^2)
Now, substitute the given values into the equations:
- Gravitational constant: G = 6.67430 × 10^-11 N m^2 / kg^2
- Mass of the sun: m_sun = 1.99 × 10^30 kg
- Mass of the earth: m_earth = 5.98 × 10^24 kg
- Mass of the moon: m_moon = 7.35 × 10^22 kg
- Distance between the sun and moon: r_SM = 1.50 × 10^11 m
- Distance between the earth and moon: r_EM = 3.85 × 10^8 m
Let's calculate the net gravitational force on the moon step by step:
1. Calculate the force due to the sun:
F_sun = (6.67430 × 10^-11 N m^2 / kg^2) * (1.99 × 10^30 kg) * (7.35 × 10^22 kg) / (1.50 × 10^11 m)^2
F_sun ≈ 4.699 × 10^20 N (approximately)
2. Calculate the force due to the earth:
F_earth = (6.67430 × 10^-11 N m^2 / kg^2) * (5.98 × 10^24 kg) * (7.35 × 10^22 kg) / (3.85 × 10^8 m)^2
F_earth ≈ 1.982 × 10^20 N (approximately)
3. Calculate the net force on the moon:
F_net = √(4.699 × 10^20 N)^2 + (1.982 × 10^20 N)^2
F_net ≈ √2.205 × 10^41 N^2
F_net ≈ 4.695 × 10^20 N (approximately)
Therefore, the magnitude of the net gravitational force on the moon is approximately 4.695 × 10^20 Newtons.