Question

Two particles each of mass 1 kg are located
at the point having co-ordinates (1, 0, 1) and
(1, 1, 0). The co-ordinates of the centre of mass
will be
​

Answer 1

Answer:

The co-ordinates of the centre of mass

will be (1,0.5,0.5)

Answer 2

Answer:

The co-ordinates of centre of mass will be $(1,\frac{1}{2} ,\frac{1}{2} ).$

Explanation:

• Concept used :

Let us consider two masses $m_{1}$ and $m_{2}$ which are located in the cartesian plane as $(x_{1} ,y_{1}, z_{1} )$ and $(x_{2} ,y_{2}, z_{2} )$ coordinates.

Then coordinates of their centre of mass will be given as $(x_{cm}, y_{cm} ,z_{zm} )$.

where as ,

• $x_{cm} =$  $x-coordinate$ of the centre of mass in the cartesian plane.

    It can be calculated as

                                        $x_{cm} = \frac{m_{1} x_{1} +m_{2} x_{2} }{m_{1}+ m_{2} }$

• $y_{cm} = y-coordinate$ of the centre of mass in the cartesian plane.
• It can be calculated as

                                          $y_{cm} = \frac{m_{1} y_{1} +m_{2} y_{2} }{m_{1}+ m_{2} }$

• $z_{cm} = z-coordinate$ of the centre of mass in the cartesian plane.

• It can be calculated as

                                           $z_{cm} = \frac{m_{1} z_{1} +m_{2} z_{2} }{m_{1}+ m_{2} }$

According to given question,

• Let mass $m_{1} = 1kg$

        Position of mass $m_{1}$ is  $({ x_{1}, y_{1} ,z_{1} }) = (1,0,1)$                    

By comparing we get,                  

$x_{1} = 1,y_{1} = 0,z_{1} =1$  

• Let mass $m_{2} = 1kg$

Position of mass $m_{2}$ is $(x_{2}, y_{2}, z_{2} ) = (1,1,0)$

By comparing we get,

        $x_{2} = 1,y_{2} = 1,z_{2} =0$

Subsituting these values in above formulas,

• $x_{cm} = \frac{m_{1} x_{1} +m_{2} x_{2} }{m_{1}+ m_{2} }$

         $x_{cm}=\frac{1* 1+1* 1}{1+ 1}\\x_{cm} = \frac{1+1}{2}\\x_{cm} = \frac{2}{2}\\x_{cm} = 1$

• $y_{cm} = \frac{m_{1} y_{1} +m_{2} y_{2} }{m_{1}+ m_{2} }$

        $y_{cm}=\frac{1* 0+1* 1}{1+ 1}\\y_{cm} = \frac{0+1}{2}\\y_{cm} = \frac{1}{2}$

• $z_{cm} = \frac{m_{1} z_{1} +m_{2} z_{2} }{m_{1}+ m_{2} }$

        $z_{cm}=\frac{1* 1+1* 0}{1+ 1}\\z_{cm} = \frac{1+0}{2}\\z_{cm} = \frac{1}{2}$

Therefore, the co-ordinates of centre of mass will be

           $(x_{cm}, y_{cm}, z_{cm} ) =$$(1,\frac{1}{2} ,\frac{1}{2} ).$

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