Question

The (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 with respect to the centre of the polygon. Find the position vector of the centre of mass.
A. \[ - \dfrac{a}{{n - 1}}\]
B. \[ - \dfrac{{{a^2}}}{{n - 1}}\]
C. \[ - \dfrac{a}{{{{\left( {n - 1} \right)}^2}}}\]
D. \[ - \dfrac{{3a}}{{2n - 1}}\]

Answer 1

Hint:In order to solve this problem, let’s see what they have given and what we need to solve. Given that we have a polygon of n side and one vacant side, we have to find the centre of mass because the rest of the masses are placed at the vertices. This can be done by using the formula of the centre of mass.


Formula Used:
The formula to find the centre of mass is given by,
\[{X_{CM}} = \dfrac{{{m_1}{x_1} + {m_2}{x_2}}}{{{m_1} + {m_2}}}\]
Where,
\[{m_1},{m_2}\] are the masses of two particles.
\[{x_1},{x_2}\] are the positions of masses.

Complete step by step solution:

Image: A regular n-polygon.

Considering the centre of a polygon as the origin with n masses. The centre of mass of a polygon with masses placed at all (n-1) corners is b. The position vectors of the centre of mass of vacant vertex be a. The formula to find the centre of mass of the given system is,
\[{X_{CM}} = \dfrac{{{m_1}{x_1} + {m_2}{x_2}}}{{{m_1} + {m_2}}} \\ \]
We can assume that if the vacant side has mass, then it will be equal to mass m. Then the centre of mass of the system comes to the origin by symmetry. Therefore, \[{X_{CM}} = 0\]
Then, \[{m_1} = \left( {n - 1} \right)m\], \[{m_2} = m\], \[{x_1} = b\] and \[{x_2} = a\]

Substituting these values in the above equation we get,
\[0 = \dfrac{{\left( {n - 1} \right)m \times b + ma}}{{\left( {n - 1} \right)m + m}} \\ \]
\[\Rightarrow 0 = \dfrac{{\left( {n - 1} \right)m \times b + ma}}{{nm}} \\ \]
\[\Rightarrow \left( {n - 1} \right)b + a = 0 \\ \]
\[\therefore b = - \dfrac{a}{{n - 1}}\]
Therefore, the position vector of the centre of mass is \[ - \dfrac{a}{{n - 1}}\].

Hence, option A is the correct answer.

Note:The centre of a polygon is a point inside a regular polygon that is equidistant from each vertex. The centre of mass of an object or system is interesting because it determines where any uniform force acting on the object will act. This is advantageous because it makes it simple to tackle mechanical issues involving the description of the motion of complex systems and objects with unusual shapes.