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NEET Physics Gravitation

Gravitation PDF Notes, Important Questions and Synopsis

SYNOPSIS

 

  • Kepler’s law:
    Law of orbit: Each planet revolves around the sun in an elliptical orbit with the sun at one of the foci.

 

Law of areas: The line joining the sun and the planet sweeps out an equal area in an equal interval of time.

Law of period: The square of the time for a planet to complete a revolution around the sun is proportional to the cube of the semi major axis of the elliptical orbit begin mathsize 12px style open parentheses straight T squared proportional to straight r cubed close parentheses end style.

  • Newton’s law of gravitation:
    Every body in the universe attracts every other body with a force which is directly proportional to the product of their masses and inversely proportional to the distance between them. 
    begin mathsize 12px style open vertical bar straight F with rightwards arrow on top close vertical bar equals fraction numerator Gm subscript 1 straight m subscript 2 over denominator straight r squared end fraction end style
  • Important characteristic of gravitational force:
    Conservative in nature
    Independent of the nature of intervening medium
    Independent of the presence or absence of the other bodies
    Central force
  •  Principle of superposition of gravitation: The resultant gravitational force F acting on a particle due to a number of point masses is equal to the vector sum of the forces exerted by the individual masses on the given particle.
    begin mathsize 12px style straight F subscript straight R equals straight F subscript 1 plus straight F subscript 2 plus. .. plus straight F subscript straight n equals sum from straight i equals 1 to straight n of straight F subscript text    end text straight i end subscript end style
  • Inertial mass and gravitational mass: Inertial mass of the body is related to its inertia of linear motion and is defined by Newton’s second law of motion.
  • Gravitational mass: Gravitational mass of a body is related to gravitational pull on the body and is defined by Newton’s law of gravitation.
  • Gravitational field:
    The space around a material body in which its gravitational pull can be experienced is called its gravitational field.
    begin mathsize 12px style straight E with rightwards arrow on top equals negative Gm over straight x squared straight x with logical and on top end style
  • Gravity:
    Force of attraction exerted by the earth towards its centre on a body lying on or near the surface of the earth. Force of gravity acting on a body is the measure of the weight of the body.

  • Relation between g and G:
    begin mathsize 12px style straight g equals GM over straight R squared end style
  • Variation of acceleration due to gravity:
    The value of acceleration due to gravity, i.e.  g changes with height, depth, shape of the earth and rotation of the earth about its own axis. It decreases with height as well as depth.
    Effect of shape: The earth is not a perfect sphere. It is flattened at the poles and bulges out at the equator; thus, the value of g is least at the equator and maximum at the poles.
  • Gravitational potential energy:
    The amount of work done in bringing the given body from infinity to that point without acceleration.
     begin mathsize 12px style straight U equals fraction numerator negative Gmm over denominator straight r end fraction end style
    Gravitational potential energy = gravitational potential mass of the body
    U = qV
  • If an isolated system consists of a particle of mass m moving with a speed v in the vicinity of a massive body of mass M, then the total mechanical energy of the particle is given by
    begin mathsize 12px style straight E equals 1 half straight m text    end text straight v squared minus fraction numerator straight G text   end text straight M text   end text straight m over denominator straight r end fraction end style           
  • Escape speed:
    The minimum speed required to project a body from the surface of the earth so that it never returns to the surface of the earth is called escape speed. Escape velocity depends on the mass and size of the planet. It is independent of the mass of the body. begin mathsize 12px style straight V subscript straight e equals square root of 2 gR subscript straight E end root end style = 11.2 km/s
  • Energy of an orbiting satellite:
     begin mathsize 12px style straight E equals fraction numerator negative Gmm over denominator 2 straight a end fraction end style
    Here, the negative sign shows that the system is bounded.
  • Geostationary or geosynchronous satellite:
    A satellite which appears to be at a fixed position at a definite height to an observer on the earth. This satellite revolves around the earth with the same angular speed in the same direction as is done by the earth around its axis.

 

FORMULAE

GRAVITATION: Universal Law of Gravitation

 begin mathsize 12px style Fα fraction numerator straight m subscript 1 straight m subscript 2 over denominator straight r squared end fraction text   end text or text   end text straight F equals fraction numerator straight m subscript 1 straight m subscript 2 over denominator straight r squared end fraction end style

Where G=6.67×10­-11Nm2Kg-2 is the universal gravitational constant.

Newton's Law of Gravitation in vector form:

 begin mathsize 12px style straight F with rightwards arrow on top subscript blank subscript 12 straight thin space end subscript end subscript equals fraction numerator Gm subscript 1 straight m subscript 2 over denominator straight r squared end fraction straight r with hat on top subscript 12 thin space    & thin space thin space straight F with rightwards arrow on top subscript blank subscript 21 straight thin space end subscript end subscript equals fraction numerator Gm subscript 1 straight m subscript 2 over denominator straight r squared end fraction straight r with hat on top subscript 21 thin space straight thin space end style

 

Now begin mathsize 12px style r with hat on top subscript 12 equals negative straight r with hat on top subscript 21 straight thin space comma straight thin space Thus thin space    straight F with rightwards arrow on top subscript 21 equals fraction numerator negative Gm subscript 1 straight m subscript 2 over denominator straight r squared end fraction straight r with hat on top subscript 12 thin space straight thin space end style

Comparing above, we get begin mathsize 12px style straight F with rightwards arrow on top subscript 12 equals negative straight F with rightwards arrow on top subscript 21 end style

Gravitational Potential: gravitational potential,

 begin mathsize 12px style straight V =− GM over straight r                    straight E equals negative dV over dr end style

  1. Ring. Error converting from MathML to accessible text. Or begin mathsize 12px style text E=− end text GMcosθ over straight x squared end style
    Gravitational field is maximum at a distance,
    r = ∓a/  and it is -2GM/3 a2
  2. Thin Circular Disc
    begin mathsize 12px style straight V equals fraction numerator negative 2 GM over denominator straight a squared end fraction open square brackets open square brackets straight a squared plus straight r squared close square brackets to the power of 1 half end exponent minus straight r close square brackets     &    straight E equals negative fraction numerator 2 GM over denominator straight a squared end fraction open square brackets 1 minus straight r over open square brackets straight r squared plus straight a squared close square brackets to the power of 1 half end exponent close square brackets equals negative fraction numerator 2 GM over denominator straight a squared end fraction open square brackets 1 minus cosθ close square brackets end style
  3.   fdfdf

    a. Point P inside the sphere. r ≤ a, then
    begin mathsize 12px style text V=− end text fraction numerator GM over denominator 2 straight a cubed end fraction open parentheses 3 straight a squared minus straight r squared close parentheses    &    straight E equals negative GMr over straight a cubed comma and space at space the space centre space straight V equals negative fraction numerator 3 GM over denominator 2 straight a end fraction straight thin space and    straight E equals 0 end style
    b. Point P outside shell.begin mathsize 12px style straight r greater or equal than straight a ,  then thin space straight V =− GM over straight r    &    straight E equals negative GM over straight r squared end style

  4. Uniform Thin Spherical Shell

    a. Point P Inside the shell
    begin mathsize 12px style straight r less or equal than straight a ,  then     straight V equals fraction numerator negative GM over denominator straight a end fraction    &    straight E equals 0 end style
    b. Point P Outside shell
    begin mathsize 12px style straight r greater or equal than straight a comma then thin space straight V equals fraction numerator negative GM over denominator straight r end fraction    &    straight E equals negative GM over straight r squared end style

    VARIATION OF ACCELERATION DUE TO GRAVITY:
  1. Effect of Altitude

    begin mathsize 12px style g subscript straight h equals GM subscript straight e over open parentheses straight R subscript straight e plus straight h close parentheses squared equals straight g open parentheses 1 plus straight h over straight R subscript straight e close parentheses to the power of negative 2 end exponent asymptotically equal to straight g open parentheses 1 minus fraction numerator 2 straight h over denominator straight R subscript straight e end fraction close parentheses when space straight h  <<  straight R. end style
  2. Effect of depth begin mathsize 12px style straight g subscript straight d equals straight g open parentheses 1 minus straight d over straight R subscript straight e close parentheses end style
  3. Effect of the surface of Earth
    The equatorial radius is about 21 km longer than its polar radius.
    We know,  begin mathsize 12px style straight g equals fraction numerator GM subscript straight e over denominator straight R subscript straight e superscript 2 end fraction Hence space straight g subscript pole space >  straight g subscript equator end style

    SATELLITE VELOCITY OR ORBITAL VELOCITY
     begin mathsize 12px style text V end text subscript 0 equals open square brackets fraction numerator GM subscript straight e over denominator open parentheses straight R subscript straight e plus straight h close parentheses end fraction close square brackets to the power of 1 half end exponent equals open square brackets fraction numerator gR subscript straight e superscript 2 over denominator open parentheses straight R subscript straight e plus straight h close parentheses end fraction close square brackets to the power of 1 half end exponent end style
    When h<<Re then begin mathsize 12px style v subscript 0 equals square root of gR subscript straight e end root end style
    Error converting from MathML to accessible text.
    Time period of Satellite
     begin mathsize 12px style straight T equals fraction numerator 2 straight pi open parentheses straight R subscript straight e plus straight h close parentheses over denominator open square brackets fraction numerator gR subscript straight e superscript 2 over denominator straight R subscript straight e plus straight h end fraction close square brackets to the power of 1 half end exponent end fraction equals fraction numerator 2 straight pi over denominator straight R subscript straight e end fraction open square brackets open parentheses straight R subscript straight e plus straight h close parentheses cubed over straight g close square brackets to the power of 1 half end exponent end style

    Energy of a Satellite

    begin mathsize 12px style text U= end text fraction numerator negative GM subscript straight e straight m over denominator straight r end fraction      straight K. straight E. equals fraction numerator GM subscript straight e straight m over denominator 2 straight r end fraction semicolon then space total space energy space rightwards arrow straight E equals negative fraction numerator GM subscript straight e straight m over denominator 2 straight R subscript straight e end fraction end style
  4. Kepler’s Laws
    Law of area:
    The line joining the sun and a planet sweeps out equal areas in equal intervals of time.
    begin mathsize 12px style table attributes columnalign left end attributes row cell Areal straight thin space velocity equals fraction numerator area space swept over denominator time end fraction equals fraction numerator 1 half straight r open parentheses rdθ close parentheses over denominator dt end fraction equals 1 half straight r squared dθ over dt equals constant end cell row cell Hence 1 half straight r squared straight omega equals constant end cell end table end style
    Law of Periods: 
    begin mathsize 12px style straight T squared over straight R cubed equals constant end style