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A non-homogenous sphere of radius R has the following density variation. rho = rho_(0), r le r//3 , rho = rho_(0)//2, (R )/(3) lt r le 3 (R )/(4), rho = (rho_(0))/(8), (3R)/(4 lt r le R, What is the gravitational field due to sphere at R = R//4 , R//2 , 5 R//6 and 2R? |
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Answer» Solution :(i) Refer to Fig. At `R = R//4`, density `= rho_(0)`. Mass of the spherical portion of the given sphere of RADIUS `R//4` is `= (4)/(3) pi ((R )/(4))^(3) rho_(0)`. Gravitational field at s DISTANCE `(R)/(4)` from the CENTRE of sphere  `I _(1) = G xx (4)/(3) pi ((R )/(4))^(3) rho_(0) xx (1)/((R//4)^(2))` `= (4)/(3) pi G(R )/(4) rho_(0) = 0.33 pi GR rho_(0)` (ii) When `r = (R )/(2)`, then for the portion of sphere of radius `R//3`, the density is `rho_(0)` and for the portion between `R//3` and `R//2`, the density is `rho_(0)//2`. So the mass of spherical portion of radius `R//2` is `= (4)/(3) pi ((R )/(3))^(3) rho_(0) + [(4)/(3) pi ((R )/(2))^(3) - (4)/(3) pi ((R )/(3))^(3)] (rho_(0))/(2)` `= (4)/(3) pi R^(3) rho_(0) [(1)/(27) + (1)/(16) - (1)/(54)]` `= (4)/(3) pi R^(3) rho_(0) xx 0.081 = 0.108 pi R^(3) rho_(0)` Gravitational field at distance `R//2` from the centre of sphere is `I_(2) = (G xx 0.108 pi R^(3) rho_(0))/((R//2)^(2)) = 0.43 pi GR rho_(0)` (ii) When `r = 5 R//6`, then for the portion of sphere of radius `R//3`, the density is `rho_(0)`, for the portion between `R//3` and `3 R//4`, the density is `rho_(0)//2` and theportion between `3 R//4` and `5 R//6`, the density is `rho_(0)//8`. Therefore, mass of the spherical portion of radius `5 R//6` is `= (4)/(3) pi ((R )/(3))^(3) rho_(0) + [(4)/(3) pi ((3R )/(4))^(3) - (4)/(3) pi ((R )/(3))^(3)] (rho_(0))/(2)` `+ [(4)/(3) pi ((5R)/(6))^(3) - (4)/(3) pi((3R)/(4))^(3)] (rho_(0))/(8)` `= 0.332 pi R^(3) rho_(0)` (on simplification) Gravitational field at a distance `5 R//6` is `I_(3) = (G xx 0.332 pi R^(3) rho_(0))/((5 R//6)^(2)) = 0.478 pi G R rho_(0)` (iv) When `r = 2 R`, then effective mass of the sphere is `= (4)/(3) pi (R//3)^(3) rho_(0) + [(4)/(3) pi ((3R )/(4))^(3) - (4)/(3) pi ((R )/(3))^(3)] (rho_(0))/(2)` `+ [(4)/(3) pi R^(3) - (4)/(3) pi ((3R)/(4))^(3)] (rho_(0))/(8)` `= 0.402 pi R^(3) rho_(0)` (On simplification) Gravitational field at disatnce `2R` is `I_(4) = (G xx 0.402 pi R^(3) rho_(0))/((2R)^(2)) =0.1 pi G R rho_(0)` |
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