CBSE Class 11th Chapter Gravitation

All objects with energy are pulled toward (or gravitate toward) one another by gravity, which is a natural phenomenon that applies to light and subatomic particles as well as stars, planets, and galaxies. Many of the structures found in the cosmos are the result of gravity, which forms hydrogen spheres and groups them into galaxies, where hydrogen fuses under pressure to make stars.It creates the tides and lends weight to tangible items on Earth. Although its impacts on items farther away become progressively less, its range is infinite.

Kepler’s Laws of Planetary Motion:
Three laws were developed by Johannes Kepler to characterize planetary motion. They are listed in the following order:

1. The orbital law: Every planet has an elliptical orbit around the sun, with the sun at one of the ellipse's foci.

2. The area law: The planet's speed fluctuates so that a vector traced from the sun to the planet's radius covers the same area in the same amount of time.

3. Law of Periods: This law is known as the Law of Periods. The square of the time period of the planet is directly proportional to the cube of the semimajor axis of its orbit.

                                                 T^2=ka^3
 where a is the semi-major axis of the elliptical orbit.

That means the time'T is directly proportional to the cube of the semi major axis, i.e., ‘a’. Let us derive the equation of Kepler’s 3rd law. Let us suppose,
m = mass of the planet
M = mass of the Sun
v = velocity in the orbit
k = constant

So, there has to be a force of gravitation between the sun and the planet.
  
                                                  F = GmM/r^2

where G is the universal gravitational constant. The value of G is 6.67 X 10-11 Nm2 kg-2 and is same throughout the universeThe dimensional formula of G is [M-1L3T-2].

Gravitational Acceleration:

Acceleration owing to gravity is the uniform acceleration caused by the earth's gravitational pull on an object falling freely.

• It has the symbol g and the unit m/s2. It points in the direction of the earth's center and is a vector quantity
• The mass of the object falling freely under gravity has no bearing on the value of g.
• The value of g varies locally by a small amount. For all intents and purposes, g is assumed to be 9.8 m/s2.
• On the moon, the acceleration caused by gravity is about. one-sixth of that is approximately 27 times more on the sun and the earth than
• Acceleration due to gravity at a height h above the surface of the earth is given by

gh = Gm / (R+h)2 = g (1 – 2h / R)

Factors Affecting Acceleration Due to Gravity

 1. Shape of Earth: Gravitational acceleration g &infi; 1 / R2 The form of Earth is elliptical. Its diameter is about 42 km smaller near the poles than it is at the equator.  As a result, g is greatest near the poles and minimum at the equator.

2. Rotation of Earth about Its Own Axis: When the earth rotates at an angle of ω with respect to its own axis, the acceleration caused by gravity at a location with a latitude of λ is determined by

g' = g – Rω2 cos2 λ

At the poles, g' = g and λ = 90°Consequently, the earth's rotation around its own axis has no influence at the poles.At the equator, g' = g – Rω2, and λ = 0°.At the equator, g has the lowest value.If earth stapes its rotation about its own axis, then g will remain unchanged at poles but increase by Rω2 at the equator.

 3. Effect of Altitude: G's value at a height of h above the surface of the earth

g' = g / (1 + h / R).2.

As a result, g drops with altitude.

4. Effect of Depth : The amount of gat depth h A from the surface of the earth

g' = g * (1 – h / R).

As a result, g drops when one descends below the earth's surface.

At the center of the earth, g = 0

Gravitational Field

The gravitational field is the area surrounding any body in which other bodies can feel its gravitational pull.

Gravitational Field Intensity:

The intensity of the gravitational field at any given position on Earth is defined as the gravitational force per unit mass.Eg or I are used to denote it.For example, I = F / m.The gravitational field's intensity at a distance r from a mass-M entity is given byFor example, I = GM / r2.It is a vector quantity pointing in the direction of the body's center of gravity.N/m is its S1 unit, while [LT-2] is its dimensional formula.The definition of gravitational mass (Mg) comes from Newton's law of gravitation.

Fg / g = Mg

Potential Gravitational:

Every location in the gravitational field has a gravitational potential equal to the work required to move a very light body from infinity to that place.The symbol for it is Vg. Potential for gravity, Vg = W / m = − GM / rIt is a scalar number with the SI unit J/kg.[L3r-2] is its dimensional formula.The gravitational potential is always negative since work W is obtained, i.e., it is negative.

Gravitational Potential Energy:

Any object's gravitational potential energy at any given location in the gravitational field is equal to the effort required to move it there from infinity. U is used to represent it.

Energy of gravitational potential U = – GMm / r

The gravitational potential energy diminishes with increasing distance, as indicated by the negative sign.Potential energy due to gravity at a height of h from the earth's surface

Uh = mgR / 1 + h/R = – GMm / R + h

Satellite: 

An object in space that orbits a planet is referred to as a satellite. Natural satellites are those celestial bodies that orbit the earth but are not created by humans. Artificial satellites are those celestial bodies that orbit the earth that are created by humans and launched for certain purposes.

Time period of satellite:

T = 2π √r3 / GM

T= 2π √(R + h)3 / g [ g = GM / R2

Near the earth's surface, time period of the satellite

T = 2π √R3 / GM = √3π / Gp

T = 2π √R / g = 5.08 * 103 s = 84 min.

where p is the average density of earth.Two categories of artificial satellites exist:

1. Parking or geostationary satellites:

A geostationary or parked satellite is one that seems to an observer on Earth to be at a fixed position at a specific height.

Elevation above Earth's surface: 36,000 kilometers

orbital radius: 42400 km

Duration: 24 Hours

Velocity of orbit: 3.1 km/s

Velocity angle = 2π / 24 = π / 12 rad / h

There, satellites orbit the planet in equatorial directions.The satellite's angular velocity about its axis is equal to that of the earth in both magnitude and direction.The aim of these satellites is communication.India's geostationary satellites are INSAT 2B and INSAT 2C.

2. Polar Satellites:

These are the satellites that orbit the planet in polar orbits. An orbit that has a 90° angle of inclination with the earth's equatorial plane is called a polar orbit.

• Elevation above Earth's surface: 880 kilometers
• Duration = 84 minutes
• Velocity in orbit = 8 km/s
• Angular velocity = 2π / 84, or π / 42 rad/min.

There, satellites orbit the planet in polar directions.These satellites are used for mapping, investigating the upper atmosphere, weather forecasting, and other purposes.India's polar satellites are part of the PSLV series.

Orbital Velocity: 

The minimal speed needed for a satellite to enter a specific orbit around the earth is known as its orbital velocity.A satellite's orbital velocity is determined by

vo = √GM / r = R √g / R + h

where M is the planet's mass, R is its radius, and h is the satellite's height above the planet's surface.If a satellite is orbiting close to the surface of the planet, then r = (R + h) = – R.orbital velocity at this pointvo = √gR

= 7.92 miles per hour

If vo is the necessary orbital velocity to move in the orbit and v is the speed of a satellite in its orbit, then

(i) The satellite will follow a parabolic route and return to Earth if v < vo.

(ii) The satellite rotates if V = vo.

Energy of a Satellite in Orbit:
total energy of a satellite

E = KE + PE
E= GMm / 2r + (- GMm / r)
E= – GMm / 2r

Binding Energy:

binding energy of the satellite is the amount of energy needed to take a satellite out of its orbit around the earth (planet) and send it into infinity.
The satellite of mass m's binding energy is provided by

BE = + GMm / 2r

Escape Velocity:  

The minimal speed at which a body must be propelled vertically upward from the surface of the earth in order to simply cross the planet's gravitational field and never return is known as Any object's escape velocity is.

ve = √2GM / R
= √2gR = √8πp GR2 / 3

The escape velocity is independent of the body's mass, form, and size as well as its projection direction.At Earth, the escape velocity is 11.2 km/s.

FAQ-

Q. What is the force of gravity?

The force of attraction between any two masses in the cosmos is called gravity. It is in charge of maintaining the orbits of celestial bodies such as planets, stars, and galaxies.

Q. How is the force of gravity calculated?

The formula to compute the gravitational force between two objects is F = (G * m1 * m2) / r^2, where m1 and m2 are the masses of the objects, r is the distance between their centers, and F is the gravitational force.

Q. What is Escape Velocity?

The minimal speed necessary for an item to break out from the gravitational pull of a celestial body, such the Earth, is known as its escape velocity. It's computed.

Q. What is free fall?
An item moving solely due to gravity is said to be in free fall. The item accelerates toward the Earth at a velocity of about 9.8 m/s2 while in free fall since there is only gravity pulling it down.