Answer:
![(x_{3},y_{3})=(-1.50\ m, -1.43\ m)](https://tex.z-dn.net/?f=%28x_%7B3%7D%2Cy_%7B3%7D%29%3D%28-1.50%5C%20m%2C%20-1.43%5C%20m%29)
Step-by-step explanation:
Given:
Mass of the first particle ![m_{1} =2.0\ kg](https://tex.z-dn.net/?f=m_%7B1%7D%20%3D2.0%5C%20kg)
Mass of the second particle ![m_{2} =4.0\ kg](https://tex.z-dn.net/?f=m_%7B2%7D%20%3D4.0%5C%20kg)
Mass of the third particle ![m_{3} =\ ?](https://tex.z-dn.net/?f=m_%7B3%7D%20%3D%5C%20%3F)
xy coordinate of the first particle ![(x_{1},y_{1})=(-1.20 m, 0.500 m)](https://tex.z-dn.net/?f=%28x_%7B1%7D%2Cy_%7B1%7D%29%3D%28-1.20%20m%2C%200.500%20m%29)
xy coordinate of the second particle ![(x_{2},y_{2})=(0.600 m, -0.750 m)](https://tex.z-dn.net/?f=%28x_%7B2%7D%2Cy_%7B2%7D%29%3D%280.600%20m%2C%20-0.750%20m%29)
coordinate of the centre of mass of three-particle system ![(x_{cm},y_{cm})=(-0.500 m, -0.700 m)](https://tex.z-dn.net/?f=%28x_%7Bcm%7D%2Cy_%7Bcm%7D%29%3D%28-0.500%20m%2C%20-0.700%20m%29)
We need to find the xy coordinate of the third particle.
Solution:
First we calculate total mass of the three-particle system.
![M = m_{1}+m_{2}+m_{3}](https://tex.z-dn.net/?f=M%20%3D%20m_%7B1%7D%2Bm_%7B2%7D%2Bm_%7B3%7D)
![M = 2.0+4.0+3.0](https://tex.z-dn.net/?f=M%20%3D%202.0%2B4.0%2B3.0)
![M = 9.0\ kg](https://tex.z-dn.net/?f=M%20%3D%209.0%5C%20kg)
x coordinate of the centre of mass of the three-particle system are given as;
![x_{cm}=\frac{1}{M}(m_{1}x_{1}+ m_{2}x_{2}+ m_{3}x_{3})](https://tex.z-dn.net/?f=x_%7Bcm%7D%3D%5Cfrac%7B1%7D%7BM%7D%28m_%7B1%7Dx_%7B1%7D%2B%20m_%7B2%7Dx_%7B2%7D%2B%20m_%7B3%7Dx_%7B3%7D%29)
Rewrite the equation for ![x_{3}](https://tex.z-dn.net/?f=x_%7B3%7D)
![m_{3}x_{3}=Mx_{cm}-m_{1}x_{1}-m_{2}x_{2}](https://tex.z-dn.net/?f=m_%7B3%7Dx_%7B3%7D%3DMx_%7Bcm%7D-m_%7B1%7Dx_%7B1%7D-m_%7B2%7Dx_%7B2%7D)
Part a.
![x_{3}=\frac{1}{m_{3}} (Mx_{cm}-m_{1}x_{1}-m_{2}x_{2})](https://tex.z-dn.net/?f=x_%7B3%7D%3D%5Cfrac%7B1%7D%7Bm_%7B3%7D%7D%20%28Mx_%7Bcm%7D-m_%7B1%7Dx_%7B1%7D-m_%7B2%7Dx_%7B2%7D%29)
Substitute all given values in above equation.
![x_{3}=\frac{1}{3.0} (9.0\times (-0.5)-2.0\times (-1.20)-4.0\times (0.6))](https://tex.z-dn.net/?f=x_%7B3%7D%3D%5Cfrac%7B1%7D%7B3.0%7D%20%289.0%5Ctimes%20%28-0.5%29-2.0%5Ctimes%20%28-1.20%29-4.0%5Ctimes%20%280.6%29%29)
![x_{3}=-1.50\ m](https://tex.z-dn.net/?f=x_%7B3%7D%3D-1.50%5C%20m)
Part b.
Similarly for
coordinate.
![y_{3}=\frac{1}{m_{3}} (My_{cm}-m_{1}y_{1}-m_{2}y_{2})](https://tex.z-dn.net/?f=y_%7B3%7D%3D%5Cfrac%7B1%7D%7Bm_%7B3%7D%7D%20%28My_%7Bcm%7D-m_%7B1%7Dy_%7B1%7D-m_%7B2%7Dy_%7B2%7D%29)
Substitute all given values in above equation.
![y_{3}=\frac{1}{3.0} (9.0\times (-0.7)-2.0\times 0.5-4.0\times (-0.75))](https://tex.z-dn.net/?f=y_%7B3%7D%3D%5Cfrac%7B1%7D%7B3.0%7D%20%289.0%5Ctimes%20%28-0.7%29-2.0%5Ctimes%200.5-4.0%5Ctimes%20%28-0.75%29%29)
![y_{3}=-1.43\ m](https://tex.z-dn.net/?f=y_%7B3%7D%3D-1.43%5C%20m)
Therefore, xy coordinate of the third particle ![(x_{3},y_{3})=(-1.50\ m, -1.43\ m)](https://tex.z-dn.net/?f=%28x_%7B3%7D%2Cy_%7B3%7D%29%3D%28-1.50%5C%20m%2C%20-1.43%5C%20m%29)