MHTCET 23 Apr 2025 Shift 1 Paper

Concept:Gravitational field due to a uniform sphere is as if its mass is concentrated at the centre; due to a thin spherical shell inside the shell is zero.Explanation:For a point at distance $x$ from the centre:• If $x < r$: field is $g = \frac{Gm}{r^3}x$ (inside solid sphere).• If $r < x < 2r$: the sphere contributes, shell contributes zero, so $g = \frac{Gm}{x^2}$.• If $x > 2r$: both masses contribute, so $g = \frac{2Gm}{x^2}$.Now check each distance:At $x = 15r$ (outside shell): $g = \frac{2Gm}{(15r)^2} = \frac{2}{225}\frac{Gm}{r^2}$, not $\frac{2}{9}\frac{Gm}{r^2}$.At $x = 2.5r$ (outside shell): $g = \frac{2Gm}{(2.5r)^2} = \frac{2}{6.25}\frac{Gm}{r^2} = \frac{8}{25}\frac{Gm}{r^2}$.At $x = 1.5r$ (between sphere and shell): $g = \frac{Gm}{(1.5r)^2} = \frac{4}{9}\frac{Gm}{r^2}$, not zero.Between the sphere and spherical shell, $g = \frac{Gm}{x^2}$ is not uniform because it depends on $x$.Answer:The correct option is B: at a distance of $2.5r$ from the centre, the gravitational field is $\frac{8}{25}\frac{Gm}{r^2}$.