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On to a sphere of radius R//2 and densit...

On to a sphere of radius `R//2` and density `P_(2)` with centre at `C_(2)` a second solid sphere is moulded with density `p_(1)` radius `R` and centre `C_(1)`. Find the force experienced by a point mass `m` at point `P` at a distance `y` from the combination as shown.

Text Solution

Verified by Experts

If we consider that a sphere of radius `R` is placed with centre at `C_(1)` od density `rho_(1)` the force on the mass at `P` is
`F_(1)=G((4//3)piR^(3)rho_(1)m)/((R+y)^(2))` towards the sphere.
If we consider that a sphere of radius `R//2` is placed with centre at `C_(2)` of density `rho_(1)` the force on the mass at `P` is
`F_(2)=G((4//3)pi(R//2)^(3)rho_(1)m)/((R//2+R+y)^(2))` towards the sphere.
If we consider that a sphere of radius `R//2` is placed with centre at `C_(2)` of density `rho_(2)` the force on the mass in at `P`
`F_(3)=(G(4//3)pi(R//2)^(3)rho_(2)m)/((R//2+R+y)^(2))`
By the principle of superposition
`F=F_(1)-F_(2)+F_(3)=(4)/(3)piR^(3)Gm[(rho_(1))/((R+y)^(2))+((rho_(2)-rho_(1))//8)/(((3R//2)+y)^(2))]`
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Knowledge Check

  • If charge q induced on outer surface of sphere of radius R, then intensity at point P at distance S from centre is

    A
    inversely proportional to `(S+R)^(2)`
    B
    inversely proportional to `R^(2)`.
    C
    inversely proportional to `S^(2)`.
    D
    directly proportional to `S^(2)`.

  • If charge q is induced on outer surface of sphere of radius R, then intensity at a point P at a distance S from its centre is

    A
    inversely proportional to `(S+R)^(2)`
    B
    inversely proportional to `R^(2)`
    C
    inversely proportional to `S^(2)`
    D
    directly proportional to `S^(2)`

  • A sphere of radius R has a charge density sigma . The electric intensity at a point at a distance r from its centre is

    A
    `E=(sigmaR^(2))/(in_(0)kr^(2))`
    B
    `E=(sigma)/(in_(0)k)`
    C
    `E=R^(2)`
    D
    `E=in_(0)kr^(2)sigma^(2)`