Gravitation handmade notes!! Aniket ncert!! 11th class

Gravitation Handmade notes by aniket ncert

Introduction

Newton’s Law of Gravitation

We have already studied the effects of gravity through the consideration of the
gravitational acceleration on earth g and the associated potential gravitational energy.

We now broaden our study and consider gravitation in a more general manner
through the

Law of Gravitation enunciated by Newton in 1687

Definition:

Every particle of matter in the universe attracts every other particle with a force that is directly proportional to the product of the masses of the particles and inversely proportional to the square of the distance between them.

Mathematically, this law, and the magnitude of the force due to the gravitational interaction between two particles, is expressed with

Fgrav = Gm1m2/r

where G = 6.67 ×10−11 N⋅m2/kg2 is the universal gravitational constant, m1 and m2

the particles masses, and r the distance between them. Equation is however not
complete since the force due to gravitation is vectorial in nature. If we define the vectors

r1 and r2 for the positions of m1 and m2 , respectively, and the unit vector that points

from r1 to r2 is er = r2 − r ( 1 ) r2 − r1 , then the force that m2 exerts on m1 is
F21 = Fgraver
= Gm1m2/r

Conversely, accordingly to Newton’s Third Law, the force that m1 exerts on m2 is
F12 = −F21
= − Gm1m2
r
2 er.
Equations taken together show that gravitation is an attractive force, i.e.,

Gravitational Interactions with Spherically Symmetric Bodies:

Although the related calculations are beyond the scope of our studies, it can be shown that gravitational interactions involving spherically symmetric bodies (e.g., uniform spheres, cavities, and shells) can be treated as if all the mass was concentrated at the centres of mass of the bodies.

This property is especially useful when dealing with the earth, other planets and satellites, as well as the sun and stars, which can all be approximated as being spherically symmetric.

There are other interesting results that can be derived for spherically symmetric bodies.

For example, the gravitational force felt by a mass m located at a radius

Kepler’s Laws

Several decades before Newton enunciated his Law of Gravitation, Johannes Kepler used
precise astronomical data of planetary motions to empirically deduce three general laws
governing the orbits of the planets orbiting the sun. These three laws are as follows:

I. Planets move on elliptical orbits about the sun with the sun at one focus.

II. The area per unit time swept out by a radius vector from the Sun to a planet is
constant.

III. The square of a planet’s orbital period is proportional to the cube of the major axis of the planet’s orbit.

These laws were all, in time, rigorously explained by Newton. But the derivation of the proof of Kepler’s First Law, in particular, requires a level of mathematical sophistication
that is beyond the scope of our studies.

An example of an elliptical planetary orbit is
shown in Figure 5, where the sun is located at the focus S . The orbit is characterized by
its semi-major axis a , which is a measure of the size of the orbit, and its eccentricity e .

The distance of the two foci from the centre of the ellipse is given by ±ea on the major
axis line. It is interesting to note that the orbits of planets in the solar system have
relatively modest eccentricities, ranging from 0.007 for Venus and 0.206 for Mercury.

The earth has an eccentricity of 0.017, while an eccentricity of zero corresponds to a
circular orbit.

The closest and furthest orbital points to the sun cross the major axis at the
perihelion and the aphelion, respectively

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Gravitation handmade notes!! Aniket ncert!! 11th class

Introduction

Newton’s Law of Gravitation

We have already studied the effects of gravity through the consideration of thegravitational acceleration on earth g and the associated potential gravitational energy.

We now broaden our study and consider gravitation in a more general mannerthrough the

Law of Gravitation enunciated by Newton in 1687

Definition:

Every particle of matter in the universe attracts every other particle with a force that is directly proportional to the product of the masses of the particles and inversely proportional to the square of the distance between them.

Mathematically, this law, and the magnitude of the force due to the gravitational interaction between two particles, is expressed with

Fgrav = Gm1m2/rwhere G = 6.67 ×10−11 N⋅m2/kg2 is the universal gravitational constant, m1 and m2

the particles masses, and r the distance between them. Equation is however notcomplete since the force due to gravitation is vectorial in nature. If we define the vectors

r1 and r2 for the positions of m1 and m2 , respectively, and the unit vector that points

from r1 to r2 is er = r2 − r ( 1 ) r2 − r1 , then the force that m2 exerts on m1 isF21 = Fgraver= Gm1m2/r

Conversely, accordingly to Newton’s Third Law, the force that m1 exerts on m2 isF12 = −F21= − Gm1m2r2 er.Equations taken together show that gravitation is an attractive force, i.e.,

Gravitational Interactions with Spherically Symmetric Bodies:

Although the related calculations are beyond the scope of our studies, it can be shown that gravitational interactions involving spherically symmetric bodies (e.g., uniform spheres, cavities, and shells) can be treated as if all the mass was concentrated at the centres of mass of the bodies.

This property is especially useful when dealing with the earth, other planets and satellites, as well as the sun and stars, which can all be approximated as being spherically symmetric.

There are other interesting results that can be derived for spherically symmetric bodies.

For example, the gravitational force felt by a mass m located at a radius

Kepler’s Laws

Several decades before Newton enunciated his Law of Gravitation, Johannes Kepler usedprecise astronomical data of planetary motions to empirically deduce three general lawsgoverning the orbits of the planets orbiting the sun. These three laws are as follows:

I. Planets move on elliptical orbits about the sun with the sun at one focus.

II. The area per unit time swept out by a radius vector from the Sun to a planet isconstant.

III. The square of a planet’s orbital period is proportional to the cube of the major axis of the planet’s orbit.

These laws were all, in time, rigorously explained by Newton. But the derivation of the proof of Kepler’s First Law, in particular, requires a level of mathematical sophisticationthat is beyond the scope of our studies.

An example of an elliptical planetary orbit isshown in Figure 5, where the sun is located at the focus S . The orbit is characterized byits semi-major axis a , which is a measure of the size of the orbit, and its eccentricity e .

The distance of the two foci from the centre of the ellipse is given by ±ea on the majoraxis line. It is interesting to note that the orbits of planets in the solar system haverelatively modest eccentricities, ranging from 0.007 for Venus and 0.206 for Mercury.

The earth has an eccentricity of 0.017, while an eccentricity of zero corresponds to acircular orbit.

The closest and furthest orbital points to the sun cross the major axis at theperihelion and the aphelion, respectively