Question

(n – 1) equal point masses each of mass m are placed at the vertices of regular n-polygon. The vacant vertex has a position vector a with respect to the centre of the polygon. Find the position vector of centre of mass.

Answer 1

The centre of mass of regular n-polygon lies at its geometric centre.

Let \(\vec{b}\) is the position vector of the centre of mass of regular n-polygon.

From questions, (n – 1) equal point masses each of mass m are placed at then (n – 1) vertices of a regular n-polygon,

Then, \(r_{cm}=\frac{(n-1)mb+ma}{(n-1)m+m}\)

Now, mass m is placed at nth remaining vertex,

then, r_cm = \(\vec{0}\)

\(\frac{(n-1)mb+ma}{(n-1)m+m}=0\)

Or \(\vec{b}=\frac{-m\vec{a}}{(n-1)m}=\frac{-\vec{a}}{(n-1)}\)

Negative sing indicated that c.m. lies other side from nth vertex geometrical centre of n-polygon.