Question

A uniform solid sphere of mass $M$ and radius $R$ is surrounded symmetrically by a uniform thin spherical shell of mass $\frac{M}{2}$ and radius $2R$. Then gravitational field at a distance of $\frac{5}{2}R$ from the centre is

• A. $\frac{2}{5}\frac{GM}{R^2}$ towards center
  
• B. $\frac{2}{5}\frac{GM}{R^2}$ away from the centre
  
• C. $\frac{6}{25}\frac{GM}{R^2}$ away from centre
  
• D. $\frac{6}{25}\frac{GM}{R^2}$ towards centre
  

Answer 1

The correct option is D $\frac{6}{25}\frac{GM}{R^2}$ towards centre



Gravitational field due to solid sphere at point $\text{P}$ is given by,

$g_1=\frac{GM}{{\left(\frac{5}{2}R\right)}^2}$

$g_1=\frac{4}{25}\frac{GM}{R^2}$

Gravitational field due to spherical shell at point $\text{P}$ is given by,

$g_2=\frac{G\left(\frac{M}{2}\right)}{{\left(\frac{5}{2}R\right)}^2}$

$g_2=\frac{2}{25}\frac{GM}{R^2}$

Both $g_1$ and $g_2$ are pointing towards the common centre of the two spheres.

Net gravitational field, $g_{net}=g_1+g_2$

$g_{net}=\frac{4}{25}\frac{GM}{R^2}+\frac{2}{25}\frac{GM}{R^2}$

$g_{net}=\frac{6}{25}\frac{GM}{R^2}$, towards the centre

Hence, option $\left(D\right)$ is the correct answer.

Why this Question? Note: Newton's Shell theorem states that, for a hollow or solid sphere, if the point under considerarion is located at any outside point, then both the spheres can be treated as point masses located at their respective centres.