(II) Every few hundred years most of the planets line up on the same side of the Sun.
Calculate the total force on the Earth due to Venus, Jupiter, and Saturn, assuming all four
planets are in a line, Fig. 5–44. The masses are mV = 0.815 mE, mj = 318 mE, mSat = 95.1
mE, and the mean distances of the four planets from the Sun are 108, 150, 778, and 1430
million km. What fraction of the Sun’s force on the Earth is this?
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To calculate the total force on Earth due to Venus, Jupiter, and Saturn, we can use the law of universal gravitation. The formula for calculating gravitational force is:
F = G * (m1 * m2) / r^2
Where:
F is the gravitational force
G is the gravitational constant (6.67 x 10^-11 N(m/kg)^2)
m1 and m2 are the masses of the two objects
r is the distance between the centers of the two objects
Since we want to calculate the force on Earth due to Venus, Jupiter, and Saturn, we'll need to calculate the force due to each planet individually and then add them up.
Let's start with Venus:
Mass of Venus (mV) = 0.815 mE
Distance from Venus to Earth (rV) = 108 million km = 108 x 10^6 km = 108 x 10^9 m
Using the formula, the force due to Venus on Earth (FV) can be calculated as:
FV = G * (mE * mV) / rV^2
Now let's move on to Jupiter:
Mass of Jupiter (mj) = 318 mE
Distance from Jupiter to Earth (rj) = 778 million km = 778 x 10^6 km = 778 x 10^9 m
The force due to Jupiter on Earth (Fj) can be calculated as:
Fj = G * (mE * mj) / rj^2
Finally, let's consider Saturn:
Mass of Saturn (mSat) = 95.1 mE
Distance from Saturn to Earth (rSat) = 1430 million km = 1430 x 10^6 km = 1430 x 10^9 m
The force due to Saturn on Earth (FSat) can be calculated as:
FSat = G * (mE * mSat) / rSat^2
To calculate the total force (FTotal) on Earth due to Venus, Jupiter, and Saturn, we'll need to add up the individual forces:
FTotal = FV + Fj + FSat
Now that we have the total force, we can calculate the fraction of the Sun's force on Earth. The force of the Sun on Earth (FSun) is simply given by the law of universal gravitation with the mass of the Sun (mSun) and the distance between the Sun and Earth (rSun).
To calculate the fraction (Frac), we can use the equation:
Frac = FTotal / FSun
Let's plug in the values and calculate the total force and the fraction of the Sun's force on Earth.
To calculate the total force on Earth due to Venus, Jupiter, and Saturn, we need to find the gravitational forces exerted by each planet individually and then sum them up.
1. Calculate the gravitational force between Earth and each planet using Newton's Law of Universal Gravitation:
F = G * (m1 * m2) / r^2
where F is the gravitational force, G is the gravitational constant (6.67430 × 10^-11 N(m/kg)^2), m1 and m2 are the masses of the two objects (in this case, Earth and the respective planet), and r is the distance between the center of masses of the two objects.
Let's calculate the gravitational forces exerted by each planet individually.
For Venus:
FV = G * (mV * mE) / rV^2
Given:
mV = 0.815 mE (mass of Venus in terms of Earth's mass)
mE = mass of Earth = 1
rV = distance between Venus and the Sun = 108 million km = 108 × 10^6 km = 108 × 10^9 m
Substituting the values:
FV = G * (0.815 * 1) / (108 × 10^9)^2
For Jupiter:
FJ = G * (mj * mE) / rJ^2
Given:
mj = 318 mE (mass of Jupiter in terms of Earth's mass)
rJ = distance between Jupiter and the Sun = 778 million km = 778 × 10^6 km = 778 × 10^9 m
Substituting the values:
FJ = G * (318 * 1) / (778 × 10^9)^2
For Saturn:
FS = G * (mSat * mE) / rS^2
Given:
mSat = 95.1 mE (mass of Saturn in terms of Earth's mass)
rS = distance between Saturn and the Sun = 1430 million km = 1430 × 10^6 km = 1430 × 10^9 m
Substituting the values:
FS = G * (95.1 * 1) / (1430 × 10^9)^2
2. Calculate the total force on Earth due to Venus, Jupiter, and Saturn by summing up the individual forces:
FTotal = FV + FJ + FS
3. Calculate the force exerted by the Sun on the Earth using the same formula:
FSun = G * (mSun * mE) / rESun^2
Given:
mSun = mass of the Sun = 1989000 mE (in terms of Earth's mass)
rESun = distance between Earth and the Sun = 150 million km = 150 × 10^6 km = 150 × 10^9 m
Substituting the values:
FSun = G * (1989000 * 1) / (150 × 10^9)^2
4. Calculate the fraction of the Sun's force on Earth exerted by Venus, Jupiter, and Saturn:
Fraction = (FTotal / FSun) * 100
Now, you can calculate the values numerically using a calculator and substitute them into the above formulas to find the total force and the fraction.