Question

Two isolated point masses m and M are separated by a distance $\ell$ . The moment of inertia of the system about an axis
passing through a point where gravitational field is zero and perpendicular to the line joining the two masses, is

• A. $\frac{m^2+M^2}{{\left(\sqrt{m}+\sqrt{M}\right)}^2}{\ell }^2$
• B. $\frac{m+M}{\left(\sqrt{m}+\sqrt{M}\right)}{\ell }^2$
• C. $\frac{m^2+M}{\left(\sqrt{m}+\sqrt{M}\right)}{\ell }^2$
• D. none of above

Answer 1

The correct option is A $\frac{m^2+M^2}{{\left(\sqrt{m}+\sqrt{M}\right)}^2}{\ell }^2$

$g=\frac{F}{m_t}$

Consider $m_t$ mass of test particle Q

$\begin{array}{c}F=\frac{G{m}_1{m}_2}{E^2}\\ \frac{m}{x^2}=\frac{M}{\left(l-x\right)}\Rightarrow \frac{M}{m}={\left(\frac{l-x}{x}\right)}^2\\ \therefore x=\frac{l}{1+\sqrt{\frac{M}{m}}}\\ I=I_m+I_M=m{x}^2+M{\left(l-x\right)}^2\\ \therefore I=\frac{m^2+M^2}{{\left(\sqrt{m}+\sqrt{M}\right)}^2}\times l^2\end{array}$