Problem

(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.S15 mE, mJ = 31S 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?

Step by step solution

Step 1</p>

First we need to calculate the masses of planets in terms of Earth’s mass (ME)

Mass of Venus

Mass of Jupiter

Mass of Saturn

Distances of planets from the Sun:

Step 2 </p>

We need to find the ratio of the gravitational force between the Earth and other planets () and the gravitational force between the Sun and Earth (

=?

Step 3</p>

First we need to calculate the gravitational force between the Earth and the planets:

The first term is due to the interaction between the Venus and the Earth, and the second one to the Earth and Jupiter and Saturn. It can be seen from the image above.

It would be easier to express everything in GME, so we will do it now.