gravitational fields

Basic definition

A gravitational field is a region of space around a mass where another mass experiences a force of attraction. It’s like an invisible web or net that pulls objects towards the source of the gravitational field.

Imagine a trampoline with a heavy ball in the center. The trampoline’s surface curves downward around the ball. If you place a smaller ball on the trampoline, it will roll towards the heavy ball because of the curved surface. This curving is analogous to how a gravitational field works in space.

Gravitational Field Strength

The strength of a gravitational field at a point in space is defined as the force per unit mass experienced by a small test mass placed at that point. Mathematically, it’s represented as:


Here's how you can calculate the gravitational field strength:

g⃗=F⃗gm \vec {g} = \frac { \vec {F}_g } {m}

Where g⃗ \vec {g} is the gravitational field strength, F⃗g \vec {F}_g is the gravitational force, and m m is the mass of the object experiencing the force.

Example

An object on Earth's surface weighs 80.0N 80.0N , and has a mass of 8.15kg 8.15kg . What is the gravitational acceleration on Earth?

Answer

g⃗=F⃗gm=80.0N8.15kg=9.82ms2 \vec {g} = \frac { \vec {F}_g } { m } = \frac { 80.0N } { 8.15kg } = 9.82 \frac {m} {s^2}

Newton's Law of Universal Gravitation

Although we learned about Newton's Law of Universal Gravitation before, let's dive deeper. It states that every point mass attracts every other point mass in the universe with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between them.


Here's the formula:

F⃗g=Gm1m2r2 \vec {F}_g = G \frac { m_1 m_2 } { r^2 }

Where G=6.67×10−11 G = 6.67 \times 10^{-11} is the gravitational constant, m1 m_1 is the mass of the object being attracted, m2 m_2 is the mass of the object attracting, and r r is the distance between them.

Example

Knowing that Earth has a mass of 5.97×1024kg 5.97 \times 10^{24} kg and a radius of 6.37×106m 6.37 \times 10^6 m , calculate the gravitational strength of Earth.

Answer

First, let's determine an equation for the gravitational strength in terms of the values we have:

F⃗g=Gm1m2r2 \vec {F}_g = G \frac { m_1 m_2 } { r^2 }

m2g⃗=Gm1m2r2 m_2 \vec {g} = G \frac { m_1 m_2 } { r^2 }

g⃗=Gm1r2 \vec {g} = G \frac { m_1 } { r^2 }


Now, all we need to do is solve:

g⃗=Gm1r2=(6.67×10−11)5.97×1024kg(6.37×106m)2=9.81ms2 \vec {g} = G \frac { m_1 } { r^2 } = (6.67 \times 10^{-11}) \frac { 5.97 \times 10^{24} kg } { (6.37 \times 10^6 m)^2 } = 9.81 \frac {m} {s^2}