Question

The masses mi are located at the points Pi. Find the moments Mx and My and the center of mass of the system. m1=4, m2=2, m3=4; P1(2, -3), P2(-3, 1), P3(3, 5)

Answer 1

Answer:

The center of mass of the system is in $P(1.4,1)$

The moments are $M_{x_{cm}}= 14 M_{y_{cm}}=10$

Step-by-step explanation:

Relevant data:

$m_{1}=4\\ m_{1}=2\\m_{1}=4\\P1(2, -3)\\P2(-3, 1)\\P3(3, 5)$

1. Calculate the Center of Mass:

The center of mass could be calculated using the equations:

$x_{cm}=\frac{m_{1}x_{1}+m_{2}x_{2}+m_{3}x_{3} }{m_{1}+m_{2}+m_{3}} \\y_{cm}=\frac{m_{1}y_{1}+m_{2}y_{2}+m_{3}y_{3} }{m_{1}+m_{2}+m_{3}}$

Then,

$x_{cm}=\frac{(4)(2)+(2)(-3)+(4)(3)}{4+2+4}=\frac{14}{10} \\y_{cm}=\frac{(4)(-3)+(2)(1)+(4)(5)}{4+2+4}}=1$

The center of mass of the system is in $P(1.4,1)$

2. Calculate the Moments

$M_{x_{cm}}=m_{1}x_{1}+m_{2}x_{2}+m_{3}x_{3} \\M_{y_{cm}}=m_{1}y_{1}+m_{2}y_{2}+m_{3}y_{3}$

Then:

$M_{x_{cm}}=(4)(2)+(2)(-3)+(4)(3)=14 \\M_{y_{cm}}=(4)(-3)+(2)(1)+(4)(5)=10$

The moments are $M_{x_{cm}}= 14 M_{y_{cm}}=10$