Three balls A,B and C of masses 50 g , 100 g and 150 g respectively are placed at the verticles of an equilateral triangle . The length of each side is 1 m . If A is placed at (0,0) and B is placed at (1,0) m , find the coordinates , (x,y) for the centres of mass of this system of balls

Options

A. `(7/12,sqrt(3/4))`m
B. `(5/18,sqrt(1/4))` m
C. `(7/12,sqrt(3/2))` m
D. `(5/18, sqrt(3/4))`m

Answer

A

Explanation

The diagram representing the given situation is shown in figure,

AP=0.5 m (`because` AC=1m `rArr AP 1/2=AC=0.5`)
From `triangle^("le")` ACP
`AC^2=AP^2+CP^2`
`1^2=0.5^2+CP^2`
`CP^2=1-0.5^2`
`CP^2=3/4`
`CP=sqrt3/2`
Coordinates of C =`(0.5, sqrt3/2)`
The centroid of a triangle is given by,
`(x,y)=((m_1x_1+m_2x_2+m_3x_3)/(m_1+m_2+m_3), (m_1y_1+m_2y_2+m_3y_3)/(m_1+m_2+m_3))`
Here,
`m_1`=50 g `(x_1,y_1)` = (0,0)
`m_2` = 100 g `(x_2,y_2)` = (1,0)
`m_3` = 150 g `(x_3,y_3)=(0.5,sqrt3/2)`
`rArr (x,y)=((50xx0+100xx1+150xx0.5)/(50xx100+150),(50xx0+100xx0+150xxsqrt3/2)/(50+100+150))`
`=((0+100+75)/300, (0+0+75sqrt3)/300)`
`=(175/300, (75sqrt3)/300)`
`=(7/12, sqrt3/4)` m