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Gravitation

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Physics Part I

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Physics

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CBSE Gujarat Board Haryana Board

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Class 10 Class 12

The earth is assumed to be a sphere of radius R. A platform is arranged at a height R from the surface of the earth. The escape velocity of a body from this platform is fve, where ve is its escape velocity from the surface of the earth. The value of f is

  • square root of 2
  • fraction numerator 1 over denominator square root of 2 end fraction
  • 1 third
  • 1 half

B.

fraction numerator 1 over denominator square root of 2 end fraction

If energy in the form of kinetic energy which is equal to binding energy, is supplied to the sphere, it leaves the gravitational field of earth.
At a platform at a height h, 
escape energy = binding energy of sphere
or space space 1 half mv subscript straight e to the power of apostrophe 2 end exponent space equals space fraction numerator GMm over denominator straight R plus straight h end fraction
or       straight v subscript straight e apostrophe space equals space square root of fraction numerator 2 GM over denominator straight R plus straight h end fraction end root space equals space square root of fraction numerator 2 GM over denominator 2 straight R end fraction end root space space left parenthesis because straight h equals straight R right parenthesis
But at surface of earth, 
                       straight v subscript straight e space equals space square root of fraction numerator 2 GM over denominator straight R end fraction end root
As given,       straight v subscript straight e apostrophe space equals space fv subscript straight e
Hence,  square root of fraction numerator 2 GM over denominator 2 straight R end fraction end root space equals space straight f square root of fraction numerator 2 GM over denominator straight R end fraction end root
    or space space space space space fraction numerator 1 over denominator 2 straight R end fraction space equals space straight f squared over straight R
therefore space space space space space straight f space equals space fraction numerator 1 over denominator square root of 2 end fraction



If energy in the form of kinetic energy which is equal to binding energy, is supplied to the sphere, it leaves the gravitational field of earth.
At a platform at a height h, 
escape energy = binding energy of sphere
or space space 1 half mv subscript straight e to the power of apostrophe 2 end exponent space equals space fraction numerator GMm over denominator straight R plus straight h end fraction
or       straight v subscript straight e apostrophe space equals space square root of fraction numerator 2 GM over denominator straight R plus straight h end fraction end root space equals space square root of fraction numerator 2 GM over denominator 2 straight R end fraction end root space space left parenthesis because straight h equals straight R right parenthesis
But at surface of earth, 
                       straight v subscript straight e space equals space square root of fraction numerator 2 GM over denominator straight R end fraction end root
As given,       straight v subscript straight e apostrophe space equals space fv subscript straight e
Hence,  square root of fraction numerator 2 GM over denominator 2 straight R end fraction end root space equals space straight f square root of fraction numerator 2 GM over denominator straight R end fraction end root
    or space space space space space fraction numerator 1 over denominator 2 straight R end fraction space equals space straight f squared over straight R
therefore space space space space space straight f space equals space fraction numerator 1 over denominator square root of 2 end fraction



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What is Heliocentric theory?

According to the Heliocentric theory, the sun is at the centre and various planets revolve around the sun at their axis. 
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What is the celestial sphere?

At night, if we see the planets and the stars in the sky, all appear to lie in the hemisphere (rest of the hemisphere we are unable to see because of being on the other side of the earth). This sphere is called the celestial sphere.
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Is Geodesic always a straight line?

No, Geodesic is a straight line if and only if,  the two points lie on the flat surface. If the two points lie on the curved surface then it is a curved line.
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The position co-ordinates of two particles of masses m1 and m2are (x1, y1, z1) and (x2, y2, z2) respectively. Find the coordinates of the centre of mass.

The position vectors of masses m1 and m2 are respectively,

space space space space space space space space space space space space r with rightwards harpoon with barb upwards on top subscript 1 equals space straight x subscript 1 straight i with overbrace on top plus straight y subscript 1 straight j with overbrace on top plus straight z subscript 1 straight k with overbrace on top

a n d space space space space space r with rightwards harpoon with barb upwards on top subscript 2 space equals space straight x subscript 2 straight i with overbrace on top plus straight y subscript 2 straight j with overbrace on top plus straight z subscript 2 straight k with overbrace on top

Let the position coordinates of the centre of mass be (X, Y, Z).

Therefore the position vector of centre of mass is,


R with rightwards harpoon with barb upwards on top equals straight X straight i with overbrace on top plus straight Y straight j with overbrace on top plus straight Z straight k with overbrace on top

Since,    R with rightwards harpoon with barb upwards on top equals fraction numerator straight m subscript 1 stack straight r subscript 1 with rightwards harpoon with barb upwards on top space plus straight m subscript 2 space straight r with rightwards harpoon with barb upwards on top subscript 2 over denominator straight m subscript 1 plus straight m subscript 2 end fraction space

therefore space space space straight X straight i with overbrace on top plus straight Y straight j with overbrace on top plus straight Z straight k with overbrace on top 

equals space space fraction numerator straight m subscript 1 left parenthesis straight x subscript 1 straight i with overbrace on top plus straight y subscript 1 straight j with overbrace on top plus straight z subscript 1 straight k with overbrace on top right parenthesis plus straight m subscript 2 left parenthesis straight x subscript 2 straight i with overbrace on top plus straight y subscript 2 straight j with overbrace on top plus straight z subscript 2 straight k with overbrace on top right parenthesis over denominator straight m subscript 1 plus straight m subscript 2 end fraction

space equals space space fraction numerator open parentheses straight m subscript 1 straight x subscript 1 plus straight m subscript 2 straight x subscript 2 close parentheses over denominator straight m subscript 1 plus straight m subscript 2 end fraction straight i with overbrace on top plus fraction numerator left parenthesis straight m subscript 1 straight y subscript 1 plus straight m subscript 2 straight y subscript 2 over denominator straight m subscript 1 plus straight m subscript 2 end fraction straight j with overbrace on topplus fraction numerator left parenthesis straight m subscript 1 straight z subscript 1 plus straight m subscript 2 straight z subscript 2 over denominator straight m subscript 1 plus straight m subscript 2 end fraction space straight k with overbrace on top

Comparing the coefficients of  straight i with overbrace on top comma space straight j with overbrace on top space and space straight k with overbrace on top, we get

straight X equals fraction numerator left parenthesis straight m subscript 1 straight x subscript 1 plus straight m subscript 2 straight x subscript 2 right parenthesis over denominator straight m subscript 1 space plus space straight m subscript 2 end fraction
straight Y space equals space straight X equals fraction numerator left parenthesis straight m subscript 1 straight y subscript 1 plus straight m subscript 2 straight y subscript 2 right parenthesis over denominator straight m subscript 1 space plus space straight m subscript 2 end fraction
straight z equals fraction numerator left parenthesis straight m subscript 1 straight z subscript 1 plus straight m subscript 2 straight z subscript 2 right parenthesis over denominator straight m subscript 1 space plus space straight m subscript 2 end fraction 


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What is Geocentric theory?

According to the geocentric theory, all the astronomical bodies like the moon, the sun and stars revolve around the earth, and the earth is at the centre of the universe. 
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