Question:
(n-1) equal point masses each of mass m are placed at the vertices of a regular n-polygon. The vacant vertex has a position vector concerning the centre of the polygon. Find the position vector of the centre of mass.
Solution:
Given,
$r_{c m}=\frac{(n-1) m b+m a}{(n-1) m+m}$
Where,
rcm is the place where mass m is placed at the nth vertex
rcm = 0
$\frac{(n-1) m b+m a}{(n-1) m+m}=0$
(n-1)mb + ma = 0
b = -ma/(n-1)m
$\vec{b}=-\frac{\vec{a}}{n-1}$
Where,
$\vec{b}$ is the position vector. Also, the negative sign indicates that the centre of mass lies on the other side of the nth vertex.