# Gravitational field – problems and solutions

1. What is the ratio of the gravitational field of the object A and the object B?

__Known :__

Mass of object A (m_{A}) = 1

Mass of object B (m_{B}) = 2

Distance of the object A from planet Earth (r_{A}) = 1

The distance of the object B from planet Earth (r_{B}) = 2

__Wanted:__ The ratio of the gravitational field of the object A and the object B

__Solution :__

The equation of the gravitational field of the Earth :

g = the magnitude of the gravitational field, G = gravitational constant, m_{B} = mass of Earth, r_{B} = radius of Earth

The ratio of the gravitational field :

g_{A} : g_{B}

G : G/2

1 : 1/2

2 : 1

2. If the position of the object A is 0.5R above the surface of the Earth, while the position of the object B is 2R above the surface of the Earth, then what is the ratio of the gravitational field that experienced by object A and object B. R is the radius of the Earth.

__Known :__

The distance of the object A from the center of the Earth = 0.5R + R = 1.5R = 1.5

The distance of the object B from the center of the Earth = 2R + R = 3R = 3

Wanted:__ The ratio of the gravitational field __

__Solution :__

The gravitational field experienced by object A (g_{A}) :

The gravitational field experienced by object B (g_{B}) :

__The ratio of ____the gravitational field ____experienced by the object A and the object B :__

3. The gravitational field at the surface of the Earth is g, then what is the gravitational field at the height of 1.5R above the surface of the Earth. R = the radius of the Earth.

__Solution__

The force of gravity (F) is directly proportional to the gravitational field (g). The greater the force of gravity, the greater the gravitational field.

The gravitational force (F) is directly proportional to 1/r^{2}, where r = the distance of an object from the center of the earth. The force of gravity is directly proportional to 1/r^{2} so that the gravitational field (g) is also proportional to 1/r^{2}.

When the astronaut is on the surface of the Earth that is R (R = radius of the earth) from the center of the earth, the gravitational field is g. This can be proven through the following calculations:

The gravitational field (g) is proportional to 1 / R^{2}. If R = 1 then 1 / R^{2} = 1/1^{2} = 1/1 = 1. If the distance of the astronaut from the center of the earth = 1 then astronaut experiences the gravitational field of = (g) (1) = g.

The distance of the astronaut from the center of the earth is 1.5R = 1.5 (1) = 1.5 then astronaut undergoes a gravitational field of : 1 / R^{2} = 1 / 1.5^{2} = 1 / 2.25. So if the astronaut distance from the center of the earth = 1.5 then astronaut has a gravitational field of = (g) (1 / 2.25) = g / 2.25

4. What is the ratio of the earth’s gravitational field for two objects, one at the surface of the earth and another at the height of ½ R from the earth’s surface (R = the radius of the earth)?

__Known :__

The distance of the object A from the center of the earth (r_{A}) = R = 1

The distance of the object B from the center of the earth (r_{B}) = 0.5R + R = 1.5R = 1.5

__Wanted:__ The ratio of the earth’s gravitational field

__Solution :__

The earth’s gravitational field experienced by object A (g_{A}) :

The earth’s gravitational field experienced by object A (g_{B}) :

The ratio of the earth’s gravitational field experienced by object A and object B :