Gravitation is the natural force of attraction that exists between all objects with mass or energy. It is one of the fundamental forces of nature and plays a significant role in shaping the structure of the universe. Gravitation is responsible for holding planets, stars, galaxies, and other celestial bodies together in space.
Gravitation is a force of attraction that exists between all objects with mass. This means that every object in the universe attracts every other object with a force that depends on their masses and the distance between them.
The Universal Law of Gravitation, formulated by Sir Isaac Newton, is a fundamental principle that describes how objects with mass interact with each other due to gravity. This law explains the force of attraction between any two objects in the universe, regardless of their size, mass, or distance.
Every object in the universe attracts every other object with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centres.
Mathematical Representation
The Universal Law of Gravitation is mathematically represented as follows:
Where:
F is the gravitational force between two objects.
G is the gravitational constant (approximately 6.67 × 10^{−11} Nm^{2} kg^{-2})
M and m are the masses of the two objects.
r is the distance between the centres of the two objects.
Key Aspects of the Law
Implications and Applications
The Universal Law of Gravitation has profound implications and is crucial for understanding various phenomena:
These characteristics are fundamental to understanding how gravitational force operates and its effects on objects in the universe. The gravitational force's ability to act over vast distances, its consistently attractive nature, and its role in shaping celestial bodies and their motions make it a crucial force in the cosmos.
The gravitational constant (G) is a proportionality constant that appears in the law of universal gravitation equations. It relates the masses of two objects to the force of gravitational attraction between them. The units of G can be derived from the units of force, distance, and mass in the equation:
Force (F) is measured in newtons (N).
Distance (r) is measured in metres (m).
Masses (M and m) are measured in kilograms (kg).
The equation for the gravitational force (F) is:
By rearranging this equation to solve for G, you get:
Substituting the units of force (N), distance (m), and masses (kg) into this equation, we get the units of G as:
G = Nm^{2} kg^{-2}
So, the SI unit of the gravitational constant (G) is Nm^{2} kg^{-2}
Value of Gravitational Constant, G:
The value of the gravitational constant (G) is approximately 6.67 × 10^{−11} Nm^{2} kg^{-2}. This value is incredibly small, which explains why the gravitational force between everyday objects is not noticeable. It takes massive objects and significant distances to observe significant gravitational forces.
Acceleration due to gravity, denoted by "g," is the acceleration that an object experiences when it falls freely under the influence of Earth's gravitational force. It's the rate at which the velocity of an object changes due to the gravitational pull as it moves downward.
Key Points
Learn more about Sources of Energy |
1. Due to the Shape of the Earth (Polar and Equatorial):
The shape of the Earth is not a perfect sphere but an oblate spheroid, slightly flattened at the poles and slightly bulging at the equator. This shape variation affects the acceleration due to gravity. The following equation expresses the ratio of the acceleration due to gravity at the poles (g_{p}) to that at the equator (g_{r}), where R_{e} is the equatorial radius of the Earth and R_{p} is the polar radius of the Earth.
Since R_{e} > R_{p}, and therefore g_{p} > g_{r}, it means that the value of g is greater at the poles and smaller at the equator due to the Earth's oblate spheroid shape.
2. Due to Altitude (Height) Above the Surface:
As you move higher above the Earth's surface, the distance between the object and the centre of the Earth increases. This leads to a decrease in the acceleration due to gravity. The following equation gives the ratio of acceleration due to gravity at height (h) above the surface (g_{h}) to that at the surface (g), where R is the Earth's radius.
As R + h > R, g_{h} > g, indicating that the value of g decreases as you move higher above the surface.
If the height h is much smaller compared to the Earth's radius (h<<R), you can use the approximation:
This formula shows that the acceleration due to gravity decreases linearly with the altitude above the Earth's surface.
3. Due to Depth Below the Surface:
As you move deeper below the Earth's surface, the distance to the centre of the Earth decreases, leading to a decrease in the acceleration due to gravity. The equation given below gives the ratio of acceleration due to gravity at depth (d) below the surface (g_{d}) to that at the surface (g).
The value of g decreases as you move deeper below the surface.
4. Due to the Rotation of the Earth (Centrifugal Effect):
The Earth's rotation causes a centrifugal force that slightly reduces the effective gravitational pull at the equator. The centrifugal acceleration is maximum at the equator and decreases as you move towards the poles so the acceleration due to gravity is maximum at the poles and minimum at the equator.
The equation given below relates the acceleration due to gravity at latitude λ (g_{λ}) to the standard acceleration due to gravity (g), where ω is the angular velocity of Earth's rotation.
At the equator (λ=0^{o}), the centrifugal effect is maximum, leading to:
g_{λequator} = g − Rω^{2}
At the poles (λ = 90^{o}), the effect is zero, so
g_{λpole} = g
The relationship between g (acceleration due to gravity) and G (gravitational constant) is described by Newton's law of universal gravitation.
From Universal Law of Gravitation:
From Newton's Second Law of Motion:
Newton's second law of motion states that the force acting on an object is equal to the mass of the object (m) multiplied by its acceleration (a):
F = m x a
The gravitational force or force of gravity can also be expressed as the weight of the object,
where g is the acceleration due to gravity.
So, we have two expressions for the force exerted by the Earth on the object.
Equating the Forces
Equating the two expressions (1 and 2) for force:
Now, we want to solve the equation for g, the acceleration due to gravity:
This equation shows the relationship between the acceleration due to gravity (g), the universal gravitational constant (G), the mass of the Earth (M), and the radius of the Earth (R).
The equations of motion for freely falling bodies are an extension of the standard equations of motion for uniformly accelerated motion, applied specifically to objects undergoing vertical motion under the influence of gravity. These equations describe how objects behave when dropped or projected upwards or downwards in the presence of gravitational force.
When an object falls vertically downward, its velocity is increasing. Thus, the acceleration due to gravity (g) is taken as positive.
g = +9.8 ms^{-2}
When an object is thrown vertically upwards, its velocity is decreasing. In this case, the acceleration due to gravity (g) is taken as negative.
g = −9.8 ms^{-2}
When an object is dropped freely from a height, its initial velocity (u) is zero.
When an object is thrown vertically upwards, its final velocity (v) becomes zero when it reaches the highest point.
The time taken by an object to rise to its highest point is equal to the time it takes to fall back from that same height.
Mass represents the amount of substance or material an object is made of. It is a scalar quantity, meaning it has magnitude but no direction.
Weight is the force with which an object is attracted toward the centre of the Earth due to gravity. Weight is not the same as mass. It represents the gravitational force exerted on an object based on its mass and the local acceleration due to gravity.
The weight of an object on the Moon refers to the force with which the Moon's gravity attracts that object towards its centre. Just like any other celestial body, the Moon has a gravitational pull, although it's weaker compared to Earth's gravity. This difference in gravitational strength between the Earth and the Moon leads to a distinct weight for an object on the Moon.
Gravity on the Moon: The Moon has a much weaker gravitational force compared to Earth. This is because the Moon has less mass and a smaller radius than Earth. The force of gravity on the Moon is approximately one-sixth (1/6) that of the force of gravity on Earth.
Calculating Weight on the Moon: The weight of an object on the Moon is calculated using the formula:
W_{m} is the weight of the object on the Moon
W is its weight on Earth
Kepler's Laws of Planetary Motion are three fundamental principles that describe how planets move in their orbits around the Sun. These laws were formulated by the German astronomer Johannes Kepler in the early 17th century and played a crucial role in advancing our understanding of celestial motion.
This law states that the path followed by a planet around the Sun is an ellipse, not a perfect circle. An ellipse is a flattened circle with two focal points (foci). The Sun is located at one of these foci.
Kepler's second law describes the speed at which a planet moves in its orbit. It states that an imaginary line connecting a planet to the Sun sweeps out equal areas in equal time intervals. This means that a planet moves faster when it's closer to the Sun (at perihelion) and slower when it's farther away (at aphelion). In other words, the orbital speed of a planet varies as it travels around the Sun, ensuring that the area it "sweeps out" remains constant in equal time intervals.
This law describes the relationship between the time it takes for a planet to complete one orbit around the Sun (its period) and the size of its orbit (measured by the semi-major axis of the ellipse). It states that the square of the orbital period (T) of a planet is directly proportional to the cube of the semi-major axis (R) of its elliptical orbit. Mathematically:
This law means that planets farther from the Sun take longer to complete their orbits, and the relationship between the two parameters is consistent across all planets.
Learn more about Our Environment |
In this section, you will find interesting and well-explained topic-wise video summary of the topic, perfect for quick revision before your Olympiad exams.
***COMING SOON***
>> Join CREST Olympiads WhatsApp Channel for latest updates.
CREST Olympiads has launched this initiative to provide free reading and practice material. In order to make this content more useful, we solicit your feedback.
Do share improvements at info@crestolympiads.com. Please mention the URL of the page and topic name with improvements needed. You may include screenshots, URLs of other sites, etc. which can help our Subject Experts to understand your suggestions easily.