Gravitation

Gravitation

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.

Universal Law of Gravitation

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.

Statement of the Law

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 Nm² 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

1. Inverse Square Law: The force of gravity weakens as the distance between the masses increases.

   

   The force is inversely proportional to the square of the distance (r²), meaning that if the distance between two objects doubles, the force of attraction becomes one-fourth. Similarly, if the distance is halved, the force becomes four times stronger.
2. Direct Proportionality: The force of gravity is directly proportional to the product of the masses of the objects.



3. Vector Nature: Gravitational force is a vector quantity, meaning it has both magnitude and direction. It acts along the line connecting the centres of the masses.
4. Symmetry: The law states that each mass attracts the other with the same force. In other words, if object A exerts a force on object B, object B exerts an equal and opposite force on object A.

Implications and Applications

The Universal Law of Gravitation has profound implications and is crucial for understanding various phenomena:

1. Celestial Orbits: The law explains the orbits of planets, moons, and other celestial bodies around each other (e.g., planets orbiting the Sun).
2. Tides: The gravitational force between the Moon, Earth, and the Sun causes ocean tides.
3. Weight: The weight of an object on the surface of a planet is the result of the gravitational force between the object and the planet.
4. Space Missions: Engineers and scientists use this law to calculate trajectories for space missions and satellite orbits.
5. Planetary Masses: By observing the orbits of planets and their moons, scientists can determine their masses.

Characteristics of Gravitation Force

1. Attractive Nature: The gravitational force is always attractive, meaning that it always pulls objects toward each other. This characteristic is evident when you drop an object – it falls to the ground due to the pull of Earth's gravity. This attraction is universal and applies to all objects with mass, regardless of their size or distance. For example, planets are attracted to the Sun, and even two small objects on Earth's surface are attracted to each other because of gravity.
2. Independent of Medium: The gravitational force is independent of the medium or material between two objects. It acts through empty space as well as through any substance, be it air, water, or any other material. This is in contrast to other forces like friction or air resistance, which can vary depending on the properties of the medium. For example, when an object falls through the atmosphere, air resistance opposes its motion, but gravity always acts downward regardless of the type of air or its density.

Conservative and Central Force

• Conservative Force: The gravitational force is a conservative force. This means that the work done by or against gravity while moving an object between two points is independent of the path taken. This characteristic leads to the concept of gravitational potential energy, where an object's position in a gravitational field determines its potential energy. As an object moves higher in the Earth's gravitational field, it gains potential energy, and as it falls, that potential energy is converted into kinetic energy.
• Central Force: The gravitational force is a central force because it acts along the line joining the centres of mass of two objects. It depends only on the masses of the two objects and the distance between them. This central nature of the force is what allows planets to move in elliptical orbits around the Sun, with the Sun at one of the foci of the ellipse.

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.

Gravitational Constant (G)

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 = Nm² kg^-2

So, the SI unit of the gravitational constant (G) is Nm² kg^-2

Value of Gravitational Constant, G:

The value of the gravitational constant (G) is approximately 6.67 × 10⁻¹¹ Nm² 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

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

1. Free Fall: When an object falls freely under the sole influence of gravity (without any other forces like air resistance), it is said to be in free fall. This concept is often demonstrated when objects, regardless of their mass, are dropped from a height and accelerate toward the Earth's surface.
2. Magnitude of Acceleration due to Gravity (g): The acceleration due to gravity near the surface of the Earth is approximately 9.8 metres per second squared (ms^-2). This means that for every second an object falls, its velocity increases by 9.8 ms^-2. This value varies slightly with location due to Earth's shape and mass distribution.
3. Calculation of g: The formula to calculate the acceleration due to gravity is derived from the law of universal gravitation and Newton's second law of motion. It can be expressed as:

   

   Where:
   G is the universal gravitational constant ( 6.67 × 10⁻¹¹ Nm² kg^-2)
   M is the mass of the Earth
   R is the distance from the object to the centre of the Earth (Earth's radius)
   
   
4. Value of g: By plugging in the values of G, M, and R for Earth, we arrive at a value of approximately 9.8 ms^-2 for the acceleration due to gravity on the Earth's surface.

Variation in the Value of g

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 − Rω²

At the poles (λ = 90^o), the effect is zero, so

g_λpole = g

Relationship between g and 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).

Equation of Motion for a Body Moving Under Gravity

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.

Modified Equations of Motion

Positive and Negative Acceleration Due to Gravity

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

Initial Velocity and Final Velocity

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.

Time of Rise and Fall

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 and Weight

Mass

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.

1. Unit: The SI unit for mass is the kilogram (kg), which is the fundamental unit for measuring the amount of matter in an object. Mass is symbolised by the lowercase letter 'm' and is also referred to as inertial mass.
2. Constant Property: An object's mass remains consistent, regardless of its location in the universe. The amount of matter in an object does not change based on where the object is located. Whether on Earth, the moon or in outer space, the mass of an object remains the same.
3. Inertia: Inertia is the tendency of an object to resist changes in its state of motion. Mass is a measure of this inertia – objects with greater mass are more resistant to changes in motion. Hence, objects with larger mass require more force to accelerate or decelerate compared to objects with smaller mass. This property is fundamental to Newton's second law of motion.
4. Zero Mass: Every object, regardless of how small, contains some amount of matter. Therefore, no object can have a mass of zero.

Weight

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.

1. Force of Attraction: Gravity is the fundamental force that gives weight to objects. The more massive an object is, the stronger the gravitational pull it experiences from Earth.
2. Formula: Weight (W) is calculated using the equation:
   W = m x g
   where 'm' is mass and 'g' is acceleration due to gravity.
3. Direction and Unit: Weight is a vector quantity with both magnitude and direction. It is measured in newtons (N). Weight has both a numerical value (magnitude) and a direction – it acts vertically downward toward the centre of the Earth.
4. Variability: Weight changes based on location due to differences in gravitational acceleration. Different locations have varying gravitational accelerations. As a result, the weight of an object on different celestial bodies or locations will differ even if its mass remains constant.
5. Weightlessness: Weight becomes zero in environments where gravitational acceleration is negligible, such as in outer space. When objects are in freefall or distant from massive bodies, they experience microgravity or weightlessness due to the very weak gravitational forces acting on them.

Weight of an Object on the Moon

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

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.

Kepler's First Law (Law of Orbits)

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 (Law of Areas)

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.

Kepler's Third Law (Law of Periods)

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.