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From a solid sphere of mass M and radius...

From a solid sphere of mass `M` and radius `R`, a solid sphere of radius `R//2` is removed as shown. Find gravitational force on mass `m` as shown

Text Solution

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Mass of smaller solid `M'=M/(4/3piR^(3)) 4/3 pi(R//2)^(2)=M/8`
Force on `m` due to sphere of mass `M` and radius `R`
`F_(1)=(GMm)/((2R)^(2))=1/4(GMm)/(R^(2))`
Force on `m` due to sphere of mass `M/8` and radius `r/2`
`F_(2)=(GM/8m)/((2R-R//2)^(2))=(GMm//8)/(9R^(2)//4)=(GMm)/(18R^(2))`
Net force on `m`
`F=F_(1)-F_(2)=(GMm)/(R^(2))(1/4-1/18)`
`=(GMm)/(R^(2))((9-2)/36)`
`=7/36(GMm)/(R^(2))`, towards left
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Knowledge Check

  • From a solid sphere of mass M and radius R, a spherical portion of radius R/2 is removed, as shown in the figure Taking gravitational potential V =0at r = oo, the potential at (G = gravitational constatn)

    A
    `(-2GM)/(3R)`
    B
    `(-2GM)/(R )`
    C
    `(-GM)/(2R)`
    D
    `(-GM)/(R )`

  • The M.I. of solid sphere of mass M and radius R about its diameter is

    A
    `(2)/(5) MR^(2)`
    B
    `(7)/(5)MR^(2)`
    C
    `(2)/(3)MR^(2)`
    D
    `(5)/(3)MR^(2)`

  • Inside a uniform sphere of mass M and radius R, a cavity of radius R//3 , is made in the sphere as shown :

    A
    Gravitational field inside the cavity is uniform
    B
    Gravitational field inside the cavity is non-uniform
    C
    The escape velocity of a particle projected from point A is `sqrt((88GM)/(15R))`
    D
    Escape velocity is defined for earth and particle system only

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