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Gravitation question

2008 · Shift 0 · Q82
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Gravitation question

2008 · Shift 0 · Q82

JEE MainPhysicsGravitationMCQ+4 / −1
This question contains Statement - 111 and Statement -222. of the four choices given after the statements, choose the one that best describes the two statements. Statement - 111: For a mass MMM kept at the center of a cube of side ′a′'a'′a′, the flux of gravitational field passing through its sides 4 π GM.4\,\pi \,GM.4πGM. Statement - 2: If the direction of a field due to a point source is radial and its dependence on the distance ′r′'r'′r′ from the source is given as 1r2,{1 \over {{r^2}}},r21​, its flux through a closed surface depends only on the strength of the source enclosed by the surface and not on the size or shape of the surface.
  1. A
    Statement - 111 is false, Statement -222 is true
  2. B
    Statement - 111 is true, Statement -222 is true; Statement -222 is a correct explanation for Statement -111
  3. C
    Statement - 111 is true, Statement -222 is true; Statement -222 is not a correct explanation for Statement -111
  4. D
    Statement - 111 is true, Statement -222 is false
View written solutionFree

Correct answer: B

  1. Use Gauss's law for gravitation

    The gravitational field due to a point mass MMM is g⃗=−GMr2 r^.\vec g = -\frac{GM}{r^2}\,\hat r.g​=−r2GM​r^.

    The gravitational flux through a closed surface is Φ=∮g⃗⋅dA⃗=−4πGM,\Phi = \oint \vec g\cdot d\vec A = -4\pi GM,Φ=∮g​⋅dA=−4πGM, if the mass MMM is enclosed.

    In many exam conventions, the magnitude of flux is written as 4πGM4\pi GM4πGM.

  2. Check Statement-1

    A mass MMM is kept at the center of a cube. The cube is a closed surface enclosing the mass.

    Hence, by Gauss's law for gravitation, Φ=−4πGM.\Phi = -4\pi GM.Φ=−4πGM. So the magnitude of flux is ∣Φ∣=4πGM.|\Phi| = 4\pi GM.∣Φ∣=4πGM.

    Therefore, Statement-1 is true in the intended JEE sense.

  3. Check Statement-2

    Statement-2 says that if a field is radial and varies as 1/r21/r^21/r2, then its flux through a closed surface depends only on the enclosed source strength, not on the size or shape of the surface.

    This is exactly the Gauss-law property for inverse-square radial fields.

    Therefore, Statement-2 is true.

  4. Does Statement-2 explain Statement-1?

    Yes. Since gravitational field of a point mass is radial and proportional to 1/r21/r^21/r2, the total flux through any closed surface enclosing the mass is independent of the shape of the surface.

    Therefore, for the cube also, the total flux is 4πGM4\pi GM4πGM in magnitude.

    So Statement-2 correctly explains Statement-1.

  5. Option matching

    Hence the correct option is: B\boxed{\text{B}}B​

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