Table of Contents
Introduction To Gravitation
Gravitation, or gravity, is a fundamental force of nature that attracts two bodies towards each other. It is the force that keeps planets in orbit around stars, moons around planets, and governs the motion of galaxies and the structure of the universe itself. The gravitational force between two masses is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers.
Newton’s Law Of Universal Gravitation
Newton’s Law Of Universal Gravitation states that every particle in the Universe attracts every other particle with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between them.
$$F = G \frac{m_{1}m_{2}}{r^{2}}$$
, where:
- $F$ is magnitude of the gravitational force between the two point masses,
- $G$ is the gravitational constant ($6.674 \times 10^{-11} \text{ Nm}^{2}\text{kg}^{-2}$),
- $m_1$is the mass of the first point mass,
- $m_2$ is the mass of the second point mass,
- $r$ is the distance between the two point masses.
The gravitational force is inversely proportional to the square of the separation of the particles (or inverse square law – more about this below).
$F \propto \frac{1}{r^{2}}$
* Only applicable to point masses (explanation on point masses provided below)
Understanding Point Masses In Gravitation
In the study of gravitation, the concept of a “point mass” is a fundamental simplification used to analyze the gravitational forces and movements between objects.
A point mass is defined as an object with mass but with negligibly small size compared to the distance between it and other objects in the system being analyzed.
- This means that the object’s dimensions are not considered in calculating gravitational forces, and the entire mass of the object is assumed to be concentrated at a central point.
- The concept of point masses is crucial in applying Newton’s Law of Universal Gravitation and analyzing gravitational fields, potential energy, and motion under gravity. It simplifies calculations by reducing the complexity of objects to their mass and distance relationships.
- For example, when considering the gravitational attraction between Earth and a falling apple, both Earth and the apple can be treated as point masses located at their respective centers of mass.
Limitations and Real-World Considerations
While the point mass approximation is widely used and generally provides accurate results for large-scale astronomical calculations and other applications, it has limitations. In situations where the size of the objects or their distance from each other is comparable, such as in calculations involving very close celestial bodies or within objects of non-uniform density, the point mass approximation may not be adequate. In these cases, more complex models that account for the distribution of mass and the shape of the objects are necessary.
The Inverse Square Law & Its Application in Newton’s Law of Universal Gravitation
The Inverse Square Law is a principle that describes the intensity of an effect such as force, light, sound, or radiation, diminishing in proportion to the square of the distance from the source of that effect. This law is foundational in various areas of physics, notably in gravitation, where it governs the strength of the gravitational force between two masses.
The Inverse Square Law can be mathematically expressed as:
$$\text{Intensity} \propto \frac{1}{r^2}$$
where $\text{Intensity}$ is the strength of the effect (e.g., gravitational force, brightness) and $r$ is the distance from the source. The law states that as the distance $r$ increases, the intensity of the effect decreases by the square of that distance. For example, if the distance doubles, the intensity of the effect becomes $\frac{1}{4}$ of its original value.
Application in Newton’s Law of Universal Gravitation
Newton’s Law of Universal Gravitation is a classic application of the Inverse Square Law. The law states that every mass attracts every other mass in the universe with a force that is inversely proportional to the square of the distance between their centers.
$$F = G \frac{m_{1}m_{2}}{r^{2}}$$
This equation illustrates that the gravitational force weakens as the square of the distance between the masses increases, in alignment with the Inverse Square Law. The law is crucial for understanding how gravitational forces operate over large distances in the universe, affecting the motion of planets, stars, and galaxies.
Worked Examples
Example 1: Basic Calculation of Gravitational Force
Two asteroids in space are 2,000 kilometers apart. Asteroid A has a mass of $2 \times 10^{12}$ kg, and Asteroid B has a mass of $3 \times 10^{12}$ kg. Calculate the gravitational force exerted between these two asteroids.
Click here to show/hide answer
Using Newton’s Law of Universal Gravitation,
$$F = G \frac{m_{1}m_{2}}{r^{2}}$$
, where $G = 6.674 \times 10^{-11} \text{ Nm}^{2}\text{kg}^{-2}$,
$m_{1} = 2 \times 10^{12}$ kg,
$m_{2} = 3 \times 10^{12}$ kg, and
$r = 2,000$ km = $2,000,000$ meters.
$$\begin{aligned} F &= 6.674 \times 10^{-11} \frac{(2 \times 10^{12}) \times (3 \times 10^{12})}{(2,000,000)^{2}} \\ &= 6.674 \times 10^{-11} \frac{6 \times 10^{24}}{4 \times 10^{12}} \\ &= 6.674 \times 10^{-11} \times 1.5 \times 10^{12} \\ &= 1.0011 \times 10^{2} \text{ N} \end{aligned}$$
Example 2: Understanding Point Masses
Why can Earth and a satellite orbiting it be considered point masses when calculating the gravitational force between them?
Click here to show/hide answer
Earth and a satellite can be considered point masses because their sizes are negligibly small compared to the distance between them. This simplification allows us to ignore their physical dimensions and treat their entire mass as if it were concentrated at their centers. This is crucial for applying Newton’s Law of Universal Gravitation simply and effectively, as it reduces the complexity involved in the calculation by focusing solely on mass and distance.
Example 3: Inverse Square Law Application
If the distance between two stars is tripled, how does the gravitational force between them change?
Click here to show/hide answer
According to the Inverse Square Law, the gravitational force between two masses is inversely proportional to the square of the distance between them. If the distance is tripled, the gravitational force becomes $\frac{1}{3^2} = \frac{1}{9}$ of its original value. This means the gravitational force is reduced to one-ninth when the distance is tripled.
Example 4: Real-World Considerations
Give an example where the point mass approximation might not be adequate for gravitational calculations and explain why.
Click here to show/hide answer
The point mass approximation might not be adequate when calculating the gravitational force between two objects that are very close to each other, such as within a galaxy where stars are relatively close, or when considering the gravitational effects inside a planet with non-uniform density. In these scenarios, the size and shape of the objects, as well as the distribution of mass, significantly affect the gravitational force, requiring more complex models that account for these factors.
Example 5: Gravitational Force and Mass Relationship
If the mass of one of two identical celestial bodies is doubled while the distance between them remains constant, how does the gravitational force between them change?
Click here to show/hide answer
When the mass of one body is doubled, the gravitational force between the two bodies, according to Newton’s Law of Universal Gravitation, becomes twice as strong because the force is directly proportional to the product of their masses. If one mass doubles, the product of the masses doubles, thereby doubling the gravitational force between them, assuming the distance remains constant.
