Question

Six objects are placed at the vertices of a regular hexagon. The geometric centre of the hexagon is at the origin with objects 1 and 4 on the $X$-axis (see figurei. The mass of the $k$ th object is $m_k=k^{i}M\mid \cos {\theta }_k\Vert$, where $i$ is an integer, $M$ is a constant with dimension of mass and ${\theta }_k$ is the angular position of the $k$ th vertex measured from the positive $X$-axis in the counter-clockwise sense. If the net gravitational force on a body at the centroid vanishes, the value of $i$ is

• A. 0
• B. 1
• C. 2
• D. 3

Answer 1

Answer: (A)

For a mass m at centroid of hexagon (at origin), net force is zero when
$\Sigma F_x=0$ and $\Sigma F_y=0$

Now, $\Sigma F_x=$ sum of all x-components of forces on m due to masses at vertices of hexagon.
$=\frac{Gm}{r^2}\left(\Sigma \left(k^{i}M\cos {\theta }_k\cos {\theta }_{k\theta }\right)\right)$
$=\frac{GmM}{r^2}(1^i\left|\cos 0^{\circ }\right|\cdot \cos 0^{\circ }+2^i\left|\cos 60^{\circ }\right|$
$\cdot \cos 60^{\circ }+3^i\left|\cos 120^{\circ }\right|\cdot \cos 120^{\circ }$
$+4^i\left|\cos 180^{\circ }\right|\cdot \cos 180^{\circ }$
$+5^i\left|\cos 240^{\circ }\right|\cdot \cos 240^{\circ }$
$+6^i\left|\cos 300^{\circ }\right|\cdot \cos 300^{\circ })$
$=\frac{GMm}{r^2}\cdot \left(1^i+\frac{2^i}{4}-\frac{3^i}{4}-4^i-\frac{5^i}{4}+\frac{6^i}{4}\right)$
As ${}^{\prime }\Sigma F_x=0,$ for net force on m to be zero, we have
$1^i+\frac{2^i}{4}-\frac{3^i}{4}-4^i-\frac{5^i}{4}+\frac{6^i}{4}=0$
Above equation is satisfied with $i=0.$