Question

A spherical hole of radius $R/2$ is excavated from the asteroid of mass $M$ as shown in the figure. The gravitational acceleration at a point on the surface of the asteroid just above the excavation is

• A. $\frac{GM}{R^2}$
• B. $\frac{GM}{2R^2}$
• C. $\frac{GM}{8R^2}$
• D. $\frac{7GM}{8R^2}$

Answer 1

The correct option is B $\frac{GM}{2R^2}$

To find the gravitational acceleration first we will have to consider that the asteroid is not excavated and find gravitational acceleration them we will assume a hypothetical sphere of negative equal to the hole and find the gravitational acceleration on its surface and subtract both:

To find mass of hole:

${\rho }_{hole}={\rho }_{asteroid}$

$\Rightarrow m_{hole}=\frac{m_{asteroid}{V}_{asteroid}}{V_{hole}}$

$=\frac{M.\frac{4}{3}\pi R^3/8}{4/3\pi R^3}$

$\Rightarrow m_{hole}=M/8$

But it will be considered as $-M/8$

$\Rightarrow a_{g_2}=\frac{-GM}{2R^2}$

Net acceleration $a_g=a_{g_1}-a_{g_2}$

$a_g=\frac{GM}{2R^2}$ towards centre of the earth