Question

The center of mass of a system of three particles of masses \[1g,2g\]and \[3g\]is taken as the origin of a coordinate system. The position vector of a fourth particle of mass \[4g\] such that the center of mass of the four particle system lies at the point \[\left( {1,2,3} \right)\] is \[\alpha \left( {\widehat i + 2\widehat j + 3\widehat k} \right)\], where \[\alpha \]is a constant . The value of \[\alpha \] is
A. \[\dfrac{5}{2}\]
B. \[\dfrac{{10}}{3}\]
C. \[\dfrac{2}{5}\]
D. \[\dfrac{1}{2}\]

Answer 1

Hint:In order to answer this question we have to find the center of mass of any one of the axes . Center of mass along x axis is calculated by
\[x = \dfrac{{{m_1}{x_1} + m{2_{}}{x_2} + {m_3}{x_3} + - - - - - - - - - {m_n}{x_n}}}{{m{1_{}} + {m_2} + {m_3} + - - - - - - - - {m_n}}}\]

Complete step-by-step solution:
Center of masses is the individual point where the weighted relative situation of the distributed mass is null.
The coordinate of x of masses \[1g,2g,3g\& 4g\] respectively-
\[\left( {{x_1} = 0,{x_2} = 0,{x_3} = 0,{x_4} = \alpha } \right)\]
Therefore, \[{x_{cm}} = \dfrac{{{m_1}{x_1} + {m_2}{x_2} + {m_3}{x_3} + - - - - - - - - - {m_n}{x_n}}}{{{m_1} + {m_2} + {m_3} + - - - - - - - - {m_n}}}\]
\[{x_{cm}} = \dfrac{{4\alpha }}{{1 + 2 + 3 + 4}}\]
Now \[1 = \dfrac{{4\alpha }}{{10}}\]
\[\therefore \alpha = \dfrac{5}{2}\]
 Therefore, option A.) \[\dfrac{5}{2}\] is the right answer.


Note:For a single rigid body, center of mass is fixed corresponding to the body and if a body has uniform thickness, it will be situated at centroid. COM of a body is discovered by the scientist Archimedes.