Gravitational field – problems and solutions

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/r², where r = the distance of an object from the center of the earth. The force of gravity is directly proportional to 1/r² so that the gravitational field (g) is also proportional to 1/r².

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 / R². If R = 1 then 1 / R² = 1/1² = 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 / R² = 1 / 1.5² = 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 :

1. What is a gravitational field?
   • Answer: A gravitational field represents the space around a mass where any other mass experiences a force due to gravity. It’s a way of describing how objects influence each other through gravitational attraction without direct contact.
2. How does the strength of a gravitational field change as you move farther from the mass producing it?
   • Answer: The strength of a gravitational field decreases with the square of the distance from the source mass. Specifically, it follows an inverse-square law: if you double the distance, the field strength becomes a quarter of its initial strength.
3. How is the gravitational field related to gravitational force?
   • Answer: The gravitational field strength at a point in space, represented by $g$, is defined as the gravitational force (F) per unit mass (m) experienced by a small test mass placed at that point. Mathematically, $g=F/m$.
4. How does the gravitational field of Earth vary with altitude?
   • Answer: As altitude increases (as you move further away from the center of the Earth), the gravitational field strength decreases. This is why astronauts in orbit experience microgravity, even though they are still within Earth’s gravitational influence.
5. What does it mean to say that gravitational fields are “conservative fields”?
   • Answer: A conservative field means that the work done on an object moving between two points in the field is independent of the path taken. For gravitational fields, this implies that the work done by the gravitational force on an object moving between two points is the same no matter which path it takes.
6. How do gravitational fields of multiple masses combine?
   • Answer: Gravitational fields from multiple masses superpose or combine vectorially. At any given point, the resultant gravitational field is the vector sum of the gravitational fields produced by each individual mass.
7. How does the mass of an object affect the gravitational field it produces?
   • Answer: The gravitational field produced by an object is directly proportional to its mass. A more massive object will produce a stronger gravitational field.
8. How is gravitational field strength measured?
   • Answer: Gravitational field strength is measured in terms of force per unit mass. In the SI unit system, it’s measured in $N/kg$ (Newtons per kilogram).
9. Can gravitational fields do work? If so, how?
   • Answer: Yes, gravitational fields can do work. When an object moves in the direction of a gravitational field (like an apple falling towards Earth), the gravitational field does work on the object, and the gravitational potential energy of the object decreases.
10. What’s the relationship between gravitational field and gravitational potential energy?

• Answer: Gravitational potential energy is the energy an object has due to its position in a gravitational field. The change in gravitational potential energy of an object is related to the work done by the gravitational force as the object moves. The gravitational field can be derived from the gradient (or spatial derivative) of the gravitational potential.