Center of Mass Calculator

Example of Few questions where you can use this Centre of mass Calculator
Question 1
Four particles of same mass 1 kg lies in x-y plane.The (x,y) coordinates of their positions are (2,2) (3,3),(-1,2) and (-1,-1) respectively. Find the position of the center of mass of the system Solution
Given $m_1=m_2 =m_3=m_4 =1$
$x_1=2$, $x_2=3$,$x_3=-1$, $x_4=-1$
$y_1=2$, $y_2=3$,$y_3=2$, $x_4=-1$
Center of Mass is calculated as
$x_{cm} =\frac {m_1x_1 + m_2x_2 + m_3x_3 + m_4x_4}{m_1 + m_2 + m_3 + m_4}= \frac {1 \times 2 + 1 \times 3 + 1 \times -1 + 1 \times -1}{1 + 1 + 1 + 1}=3/4$
$y_{cm} =\frac {m_1y_1 + m_2y_2 + m_3y_3+ m_4y_4}{m_1 + m_2 + m_3 + m_4}=\frac {1 \times 2 + 1 \times 3 + 1 \times 2 + 1 \times -1}{1 + 1 + 1 + 1}=3/2$

Question 2
Three particles of masses 1 kg,2 kg and 3 kg respectively lies in x-y-z plane.The (x,y,z) coordinates of their positions are (2,2,1) (3,3,1),(-1,2,3) respectively. Find the position of the center of mass of the system
Solution
Given $m_1=1 \ kg,m_2 =2 \ kg ,m_3= 3 \ kg$
$x_1=2$, $x_2=3$,$x_3=-1$
$y_1=2$, $y_2=3$,$y_3=2$
$z_1=1$, $z_2=1$,$z_3=3$
Center of Mass is calculated as
$x_{cm} =\frac {m_1x_1 + m_2x_2 + m_3x_3 }{m_1 + m_2 + m_3 }= \frac {1 \times 2 + 2 \times 3 + 3 \times -1 }{1 + 2 + 3 }=5/6$
$y_{cm} =\frac {m_1y_1 + m_2y_2 + m_3y_3}{m_1 + m_2 + m_3 }=\frac {1 \times 2 + 2 \times 3 + 3 \times 2 }{1 + 2 + 3 }=15/6= 5/2$
$z_{cm} =\frac {m_1z_1 + m_2z_2 + m_3z_3}{m_1 + m_2 + m_3 }=\frac {1 \times 1 + 2 \times 1 + 3 \times 3 }{1 + 2 + 3 }=12/6= 2$