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 a with respect to the centre of the polygon. Find the position vector of centre of mass.

Answer 1

The centre of mass of a regular polygon with n sides lies on its geometric centre. If mass m is placed at all the n vertices, then the C.O.M Is again at the geometric centre. Let `vecr` be the position vector of the COM and `vecA` of the vacant vertex. Then

`r_cm = ((n - 1)mr + ma)/((n - 1)m + m)` = 0  .....(when mass is placed at n^th vertex also)

`(n - 1)mr + ma` = 0

`r = - (ma)/((n - 1)m)`

`vecr = - veca/((n - 1))`

The negative sign depicts that the C.O.M lies on the opposite side of n^th vertex.

Hence the center of mass of n particles is a weighted average of the position vectors of n particles making up the system.

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

Let the position vector of each centre of mass or regular n polygon is r.

(n – 1) equal point masses each of mass m are placed at (n – 1) vertices of the regular n-polygon, therefore, for its centre of mass.