Question

A thin hemispherical shell of mass $M$ and radius $R$ is placed as shown in the figure The magnitude of gravitational field at P due to thin hemispherical shell is $l_0$ The magnitude of gravitational field at 'Q' due to thin hemispherical shell is given by

• A. $I_0/2$
• B. $I_0$
• C. $\frac{2GM}{9R^2}-I_0$
• D. ${\frac{2GM}{9R}}^2+I_0$

Answer 1

Answer: (C)

$\frac{2GM}{9R^2}-I_0$

The gravitational field at a distance $r$ , due to a spherical shell of mass $M$ is given by ,

$E_g=\frac{GM}{r^2}$

Let we consider that the spherical shell is made of two hemispheres (upper and lower) each of mass M,

Therefore gravitational field at a distance $3R$ (point $Q$) , due to spherical shell of mass $2M$ will be ,

$E_g=\frac{2GM}{(3R{)}^2}=\frac{2GM}{9R^2}$ ...................$\left(1\right)$

and it is given that the gravitational field at a distance of $3R$ from hemispherical shell is $I_o$ (as at point P). Now if we remove lower hemispherical shell then total gravitational field given by eq.1 will be reduced by $I_o$ (gravitational field due to lower hemispherical shell) at point $Q$ i.e.

$E_g^{\prime }=\frac{2GM}{9R^2}-I_o$