Consider the earth following its nearly circular orbit (dashed curve) about the sun.(Figure 2) The earth has mass mearth=5.98×1024kg and the sun has mass msun=1.99×1030kg. They are separated, center to center, by r=93millionmiles=150millionkm.
To calculate the gravitational force between the Earth and the Sun, we can use Newton's law of universal gravitation, which states that the force between two objects is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers.
The formula for the gravitational force (F) between two objects is:
F = (G * m1 * m2) / r^2
Where:
- F is the gravitational force between the two objects
- G is the gravitational constant (G ≈ 6.67430 × 10^-11 N m^2 / kg^2)
- m1 is the mass of the first object (in this case, the Sun)
- m2 is the mass of the second object (in this case, the Earth)
- r is the distance between the centers of the two objects (150 million km or 93 million miles in this case)
Let's calculate the gravitational force:
m1 (mass of the Sun) = 1.99 × 10^30 kg
m2 (mass of the Earth) = 5.98 × 10^24 kg
r = 150 × 10^6 km = 150 × 10^6 × 10^3 = 1.5 × 10^11 m
Plugging these values into the formula:
F = (G * m1 * m2) / r^2
F = (6.67430 × 10^-11 N m^2 / kg^2) * (1.99 × 10^30 kg) * (5.98 × 10^24 kg) / (1.5 × 10^11 m)^2
Now, let's calculate the value of the gravitational force.
To calculate the gravitational force between the Earth and the Sun, you can use Newton's law of gravitation:
F = G * (m1 * m2) / r^2
Where:
F is the gravitational force between the Earth and the Sun,
G is the gravitational constant (approximately 6.674 × 10^-11 N m^2/kg^2),
m1 is the mass of the Earth,
m2 is the mass of the Sun, and
r is the distance between the centers of the Earth and the Sun.
Let's substitute the given values into the formula:
F = (6.674 × 10^-11 N m^2/kg^2) * ((5.98 × 10^24 kg) * (1.99 × 10^30 kg)) / (1.5 × 10^11 m)^2
Now, let's solve this equation step by step:
1. Convert 93 million miles to kilometers:
1 mile = 1.60934 kilometers
93 million miles = 93 million * 1.60934 kilometers = 149.6 million kilometers
2. Substitute the values into the formula:
F = (6.674 × 10^-11 N m^2/kg^2) * ((5.98 × 10^24 kg) * (1.99 × 10^30 kg)) / (1.496 × 10^11 m)^2
3. Simplify the equation:
F = (6.674 × 10^-11 N m^2/kg^2) * (1.19 × 10^55 kg^2) / (2.2416 × 10^22 m^2)
4. Calculate the value of F:
F = 2.985 × 10^44 N
Therefore, the gravitational force between the Earth and the Sun is approximately 2.985 × 10^44 Newtons.