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gravitation

Gravitation

Definition: Gravitation is the force of attraction between any two objects in the universe that have mass.

Examples:

An apple falling towards the Earth.
The Moon revolving around the Earth.
Planets revolving around the Sun.
Objects remaining on the Earth's surface due to gravity.

Gravity and Gravitation

Gravitation: The force of attraction between any two objects having mass.
Gravity: The gravitational force exerted by a celestial body, such as the Earth, on objects near it.

Circular Motion and Centripetal Force

Circular Motion

The motion of an object along a circular path is called circular motion.

Examples:

A stone tied to a string and rotated in a circle.
The motion of the Moon around the Earth.
The motion of a satellite around the Earth.

Uniform Circular Motion

When an object moves along a circular path with constant speed, its motion is called uniform circular motion.

Although its speed remains constant, its velocity changes because its direction changes continuously.

Centripetal Force

Definition: The force acting on an object moving in a circular path and directed towards the centre of the circle is called centripetal force.

Formula:

F = mv²/r

Where:

F = Centripetal force
m = Mass of the object
v = Speed of the object
r = Radius of the circular path

SI unit: Newton (N)

Kepler's Laws of Planetary Motion

Johannes Kepler formulated three laws describing the motion of planets around the Sun.

First Law: Law of Orbits

Every planet revolves around the Sun in an elliptical orbit, with the Sun at one of the two foci.

Second Law: Law of Areas

The line joining a planet and the Sun sweeps out equal areas in equal intervals of time.

Third Law: Law of Periods

The square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit.

Formula:

T² ∝ r³

Where:

T = Orbital period
r = Semi-major axis of the orbit

For planets orbiting the same central body:

T₁²/T₂² = r₁³/r₂³

These laws helped scientists understand planetary motion and the role of gravitational force.

Newton's Universal Law of Gravitation

Statement

Every object in the universe attracts every other object with a force that is directly proportional to the product of their masses & inversely proportional to the square of the distance between their centres.

Mathematical Formula

F = Gm₁m₂/r²

Where:

F = Gravitational force
G = Universal gravitational constant
m₁ and m₂ = Masses of the two objects
r = Distance between their centres

The value of the universal gravitational constant is approximately:

G = 6.67 × 10⁻¹¹ N m²/kg²

Important Properties

Gravitational force is always attractive.
It acts between all objects having mass.
It increases when either mass increases.
It decreases as the distance between objects increases.
It acts along the line joining the centres of the two objects.

Inverse Square Law

Gravitational force is inversely proportional to the square of the distance between two objects.

F ∝ 1/r²

If the distance between two objects doubles, the gravitational force becomes one-fourth of its original value.

If the distance becomes three times the original distance, the force becomes one-ninth of its original value.

Acceleration Due to Gravity

Definition: The acceleration produced in a freely falling object due to the Earth's gravitational force is called acceleration due to gravity.

It is represented by the symbol g.

Near the Earth's surface:

g ≈ 9.8 m/s²

The direction of acceleration due to gravity is towards the centre of the Earth.

Formula for Acceleration Due to Gravity

g = GM/R²

Where:

G = Universal gravitational constant
M = Mass of the Earth
R = Radius of the Earth

Relation Between G and g

The gravitational force acting on an object of mass m near the Earth's surface is:

F = GMm/R²

According to Newton's second law:

F = mg

Equating the two expressions:

mg = GMm/R²

Cancelling m:

g = GM/R²

This equation shows that acceleration due to gravity depends on the mass and radius of the Earth.

Mass & Weight

Mass

Mass is the quantity of matter contained in an object.

Symbol: m
SI unit: kilogram (kg)
It remains approximately constant from one location to another.
It is a scalar quantity.

Weight

Weight is the gravitational force acting on an object.

Formula:

W = mg

Where:

W = Weight
m = Mass
g = Acceleration due to gravity

SI unit: Newton (N)

Weight depends on the local gravitational acceleration, so it can vary from one celestial body to another.

Free Fall

Definition: When an object moves under the influence of gravity alone, its motion is called free fall.

In the absence of air resistance, all objects near the Earth's surface have the same acceleration due to gravity, regardless of their masses.

Equations of Motion for Free Fall

For downward motion, taking downward as positive:

v = u + gt

s = ut + ½gt²

v² = u² + 2gs

Where:

u = Initial velocity
v = Final velocity
g = Acceleration due to gravity
t = Time
s = Displacement

For an object released from rest, u = 0.

Therefore:

v = gt

s = ½gt²

v² = 2gs

Motion of an Object Thrown Vertically Upwards

When an object is thrown vertically upwards, gravity acts downward and reduces its upward velocity.

Taking upward as positive:

v = u − gt

h = ut − ½gt²

v² = u² − 2gh

At the maximum height, the final velocity becomes zero.

Therefore:

Maximum height: h = u²/(2g)

Time to reach maximum height: t = u/g

Ignoring air resistance, the time taken to rise to the maximum height equals the time taken to return to the original level.

Escape Velocity

Definition: Escape velocity is the minimum initial speed required for an object to escape the gravitational field of a celestial body without further propulsion, assuming no air resistance.

Formula:

vₑ = √(2GM/R)

For the Earth, the escape velocity from its surface is approximately 11.2 km/s under the idealized model.

Where:

vₑ = Escape velocity
G = Universal gravitational constant
M = Mass of the Earth
R = Radius of the Earth

Escape velocity depends on the mass and radius of the celestial body.

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