Question

A uniform sphere of mass M and radius R exerts a force F on a small mass m situated at a distance of 2R from the centre O of the sphere. A spherical portion of diameter R is cut from the sphere as shown in the figure. The force of attraction between the remaining part of the sphere and the mass m will be

• A. $\frac{7F}{9}$
• B. $\frac{2F}{3}$
• C. $\frac{4F}{9}$
• D. $\frac{F}{3}$

Answer 1

The correct option is A. $\frac{7F}{9}$.

The force of attraction between the complete sphere and mass m is
$F=\frac{GmM}{(2R{)}^2}=\frac{GmM}{4R^2}$
Mass of complete sphere is $M=\frac{4\pi }{3}{R}^3\rho$, where $\rho$ is the density of the sphere.
Mass of the cut-out portion is $m_0=\frac{4\pi }{3}{\left(\frac{R}{2}\right)}^3\rho =\frac{M}{8}$.
Now, the distance between the centre of the cut out portion and mass m $=2R-\frac{R}{2}=\frac{3R}{2}$
Hence, the force of attraction between the cut out portion and mass m is $f=\frac{G{m}_{0}m}{{\left(\frac{3R}{2}\right)}^2}=\frac{G\left(\frac{M}{8}\right)m}{\frac{9R^2}{4}}=\frac{GmM}{4R^2}\times \frac{2}{9}=\frac{2F}{9}$
Therefore, the force of attraction between the remaining part of the sphere and mass m $=F-f=F-\frac{2F}{9}=\frac{7F}{9}$ which is choice (a).