Question

The magnitudes of the gravitational potentials at distances r₁ and r₂ from the centre of a uniform sphere of radius R and mass M are V₁ and V₂ respectively. Then ______.

विकल्प

• `"V"_1/"V"_2 = "r"_1^2/"r"_2^2` ; if r₁ < R and r₂ < R.

• `"V"_1/"V"_2 = "r"_2^2/"r"_1^2` ; if r₁ > R and r₂ > R.

• `"V"_1/"V"_2 = "r"_2/"r"_1` ; if r₁ < R and r₂ < R.

• `"V"_1/"V"_2 = "r"_1/"r"_2` ; if r₁ > R and r₂ > R.

Answer 1

The magnitudes of the gravitational potentials at distances r₁ and r₂ from the centre of a uniform sphere of radius R and mass M are V₁ and V₂ respectively. Then `"V"_1/"V"_2 = "r"_1^2/"r"_2^2` ; if r₁ < R and r₂ < R.

Explanation:

For r₁ > R and r₂ > R, Potential at r is,

`"V" = "GM"/"r"`

here, M = `4/3pi "R"^3"p"`

`therefore "V"_1 = (4/3pi "R"^3"pG")/"r"_1  "and"  (4/3pi "R"^3"pG")/"r"_2`

`therefore "V"_1/"V"_2 =  "r"_2/"r"_1`

For r₁ < R and r₂ < R,

The gravitational potential inside the earth is only due. to the mass of the earth that lies within a solid sphere of radius r.

`therefore "M" = 4/3pi "r"^3"p"`

`therefore "V"_1 = (4/3pi "r"_1^3"pG")/"r"_1 = 4/3pi "r"_1^2"pG"`

`therefore "V"_2 = (4/3pi "r"_2^3"pG")/"r"_2 = 4/3pi "r"_2^2"pG"`

`therefore  "V"_1/ "V"_2 = "r"_1^2/"r"_2^2`