# Every few hundred years most of the planets line up on the same side of the sun.

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 line. The masses are Mv=.815Me, Mj=318Me, Ms=95.1Me, and their mean distances from the sun are 108, 150, 778, and 1430 million km, respectively. What fraction of the sun's force on the Earth is this?

## General guidance

Concepts and reason

The concepts used to solve this problem are Newton’s law of Gravitation and vector addition in one dimension.

Use the expression for gravitational force between two objects separated by a distance to get the force between earth and other planets.

Use the expression for vector addition to get the net force on earth by the three planets.

Use the expression for gravitational force between two objects separated by a distance to get the force between earth and sun.

Use the expression for division to get what fraction of sun’s force is the net force on earth by the three planets.

Fundamentals

The expression for gravitational force between two objects is,

Here, the universal gravitational constant is, mass of object 1 is , the mass of the 2nd object is and the separation between the two objects is .

The expression for magnitude of vector addition in one dimension, when the vectors are parallel is,

Here, the net force is, the two forces are and .

The expression for magnitude of vector addition in one dimension, when the vectors are antiparallel is,

## Step-by-step

### Step 1 of 3

(a)

The expression for gravitational force between two objects is,

The distance between sun and the Venus is .

The distance between sun and the Earth is .

The distance between sun and the Jupiter is .

The distance between sun and the Venus is .

The distance between sun and the Saturn is .

The distance between Earth and Venus is,

Substitute for and for .

The distance between Earth and Jupiter is,

Substitute for and for .

The distance between Earth and Saturn is,

Substitute for and for .

The gravitational force between Earth and Venus is,

The gravitational force between Earth and Jupiter is,

The gravitational force between Earth and Saturn is,

The gravitational force between two massive objects is directly proportional to the mass of the objects and inversely proportional to the square of the distance between them. This is the statement of Newton’s gravitational law.

### Step 2 of 3

All the planets are in a straight line and hence the force of Venus on earth will be in one direction and the two forces due to Jupiter and Saturn will be in the opposite direction.

So, the net gravitational force on earth due to the three planets is,

Substitute for , for and for .

Again, substitute for , for , for , for , for , for, for and for .

After calculation,

Part a

The net force on earth due to the three planets is .

The problem is a one-dimensional problem and the force vectors are either parallel or antiparallel to each other. While adding the vectors, the usual way of vector addition is not required.

### Step 3 of 3

(b)

The expression for gravitational force between earth and sun is,

Here, the mass of earth is , the mass of the sun is , the distance between sun and earth is .

Substitute for , for and for and for .

The net force on earth by the three planets is a fraction of sun’s force on earth. To get it, divide the force between sun and earth by the net force on earth due to the three planets.

Part b

The force on earth due to the three planets is fraction of the force between earth and sun.

The force on earth by the three planets is extremely small compared to the force exerted by sun on earth. Gravitational force depends on the mass. As the mass of sun is huge compared to the planets, its gravitational pull is too stronger.

Part a

The net force on earth due to the three planets is .

Part b

The force on earth due to the three planets is fraction of the force between earth and sun.

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