Calculating Gravitational Force
Newton's law of universal gravitation states that every body of nonzero mass attracts every other object in the universe. This attractive force is called gravity. It exists between all objects, even though it may seem ridiculous.
For example, while you read these words, a tiny force arises between you and the computer screen. This force is too small to cause any visible effect, but if you apply the principle of gravitational force to planets or stars, its effects will begin to show.
What is the gravity equation?
Use the following formula to calculate the gravitational force between any two objects:
F = GMm/R²
where:
- F — Gravitational force, measured in newtons (N) . It is always positive, which means that two objects of a certain mass always attract (and never repel) each other;
- M and m — Masses of two objects in question, in kilograms (kg);
- R — Distance between the centers of these two objects, in meters (m); and
- G — Gravitational constant. It is equal to 6.674×10⁻¹¹ N·m²/kg².
The attractive nature of gravity is also connected with a new definition of mass. The theory of General Relativity, introduced by Albert Einstein in 1915, establishes that curvature = mass, which means that any massive object — including the Sun, the Earth, and you — deforms or curves the spacetime, creating a gravity well.
Example
Calculating Gravitational Force
The gravitational force refers to the attractive force between two objects with mass. It depends on the masses of the objects and the distance between them. According to Newton's law of universal gravitation, this force is always attractive and acts along the line joining the two objects' centers.
The general approach to calculating gravitational force includes:
- Identifying the two objects and their respective masses.
- Knowing the distance between the centers of the two objects.
- Applying the gravitational force formula to calculate the force between the objects.
Gravitational Force Formula
The general formula for gravitational force is:
\[ F = \frac{G M_1 M_2}{r^2} \]
Where:
- F is the gravitational force (in newtons, N).
- G is the gravitational constant, \( G = 6.674 \times 10^{-11} \, \text{N} \cdot \text{m}^2/\text{kg}^2 \).
- M₁ and M₂ are the masses of the two objects (in kg).
- r is the distance between the centers of the two objects (in meters).
Example:
If two objects, each with a mass of \( 5 \times 10^{24} \, \text{kg} \), are separated by a distance of \( 6.4 \times 10^6 \, \text{m} \), the gravitational force between them is:
- Step 1: Use the gravitational force formula: \( F = \frac{{(6.674 \times 10^{-11})(5 \times 10^{24})(5 \times 10^{24})}}{{(6.4 \times 10^6)^2}} \).
- Step 2: Calculate the result: \( F = 1.30 \times 10^{21} \, \text{N} \).
Gravitational Force for Different Objects
The gravitational force varies based on the masses of the objects and the distance between them. For example, the force between Earth and the Moon is approximately \( 1.98 \times 10^{20} \, \text{N} \), while the force between Earth and a satellite is much weaker due to the distance involved.
Example:
If the mass of the Earth is \( 5.97 \times 10^{24} \, \text{kg} \) and the mass of the Moon is \( 7.35 \times 10^{22} \, \text{kg} \), with a distance of \( 3.84 \times 10^8 \, \text{m} \), the gravitational force between the Earth and the Moon is approximately \( 1.98 \times 10^{20} \, \text{N} \), calculated using the gravitational force formula.
Real-life Applications of Gravitational Force
Gravitational force plays a crucial role in various fields such as:
- The movement of celestial bodies in space (planets, moons, asteroids, etc.).
- Space missions that involve sending spacecraft and satellites into orbit.
- Understanding the effects of gravity on tides and ocean currents on Earth.
Common Units for Gravitational Force
SI Unit: The unit of gravitational force is newtons (N).
Gravitational force depends on the masses of the objects and the distance between them, and it is always attractive.
Common Operations with Gravitational Force
Force Between Different Celestial Bodies: The gravitational force changes based on the masses and the distance between the objects.
Force Between Objects on Earth: The gravitational force between objects on Earth is what gives weight to the objects. The force of gravity is commonly represented as \( F = mg \), where \( g \) is the acceleration due to gravity on Earth.
Effects of Gravity in Space: Gravitational force keeps planets in orbit around stars, moons around planets, and it influences the paths of comets and asteroids.
| Problem Type | Description | Steps to Solve | Example |
|---|---|---|---|
| Gravitational Force Between Two Planets | Finding the gravitational force between two planets based on their masses and the distance between them. |
|
If the mass of the first planet is \( 5 \times 10^{24} \, \text{kg} \), the mass of the second planet is \( 3 \times 10^{23} \, \text{kg} \), and the distance between them is \( 4 \times 10^7 \, \text{m} \), the gravitational force is \( F = \frac{(6.674 \times 10^{-11})(5 \times 10^{24})(3 \times 10^{23})}{(4 \times 10^7)^2} = 6.26 \times 10^{17} \, \text{N} \). |
| Gravitational Force Between Earth and an Object | Finding the gravitational force acting on an object near Earth's surface. |
|
If the mass of the object is \( 100 \, \text{kg} \) and the mass of Earth is \( 5.97 \times 10^{24} \, \text{kg} \) with a distance of \( 6.38 \times 10^6 \, \text{m} \), the gravitational force is \( F = \frac{(6.674 \times 10^{-11})(100)(5.97 \times 10^{24})}{(6.38 \times 10^6)^2} = 980 \, \text{N} \). |
| Gravitational Force on the Moon | Finding the gravitational force between the Earth and the Moon. |
|
If the mass of Earth is \( 5.97 \times 10^{24} \, \text{kg} \), the mass of the Moon is \( 7.35 \times 10^{22} \, \text{kg} \), and the distance between them is \( 3.84 \times 10^8 \, \text{m} \), the gravitational force is \( F = \frac{(6.674 \times 10^{-11})(5.97 \times 10^{24})(7.35 \times 10^{22})}{(3.84 \times 10^8)^2} = 1.98 \times 10^{20} \, \text{N} \). |
| Gravitational Force Near a Black Hole | Finding the gravitational force near a black hole. |
|
If the mass of the black hole is \( 10^6 \, \text{M}_\odot \) (solar masses), the distance from the black hole is \( 1 \times 10^4 \, \text{m} \), and the mass of the object is \( 10^2 \, \text{kg} \), the gravitational force is \( F = \frac{(6.674 \times 10^{-11})(10^6)(1.99 \times 10^{30})(10^2)}{(1 \times 10^4)^2} = 1.33 \times 10^{16} \, \text{N} \). |
