Question

A uniform metal sphere of radius $\left(a\right)$ and mass $\left(m\right)$ is surrounded by a thin, uniform spherical shell of equal mass and radius $\left(4a\right)$. The centre of the shell falls on the surface of the inner sphere. Find the gravitational field ratio at the points$P_{1}and{P}_2$shown in figure:

• A. $\frac{16}{900}$
• B. $\frac{61}{900}$
• C. $\frac{900}{61\times 16}$
• D. None of these

Answer 1

The correct option is C $\frac{900}{61\times 16}$

As we know that the gravitational field is given by,

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

At point $P_1$ field will be due to inner solid sphere.

Gravitational field at $P_1$,

$E=\frac{GM}{(3a+a{)}^2}=\frac{GM}{16a^2}$

At point $P_2$ field will be due to both the spheres.

I.e. Gravitational field at $P_2,E=E_1+E_2$

$=\frac{GM}{(a+4a+a{)}^2}+\frac{GM}{(4a+a{)}^2}$

$=\frac{GM}{a^2}(\frac{1}{36}+\frac{1}{25})$

$=\left(\frac{61}{900}\right)\frac{GM}{a^2}$

Ratio of gravitational field$P_1$ and $P_2$,

$\frac{P_1}{P_2}=\frac{\frac{GM}{16a^2}}{\frac{61GM}{900a^2}}$

$\frac{P_1}{P_2}=\frac{900}{16\times 61}$