Four particles are in a 2-D plane with masses, x- and y- positions, and x- and y

Question

Four particles are in a 2-D plane with masses, x- and y- positions, and x- and y- velocities as given in the table below:

3) What is the speed of the center of mass?

4) When a fifth mass is placed at the origin, what happens to the horizontal (x) location of the center of mass?

5) When a fifth mass is placed at the center of mass, what happens to the vertical (y) location of the center of mass?

m x (meters) y (meters) vx (m/s) vy (m/s) 1 8.3 kg -2.7 -4.4 3.2 -3.9 2 8.5 kg -3.7 3.4 -5.2 4.9 3 8.6 kg -4.7 5.5 -6.2 1.9 4 8.8 kg 5.4 2.5 4.2 -3

Explanation / Answer

m1 = 8.3
m2 = 8.5
m3 = 8.6
m4 = 8.8

Vy of the centre of mass = ( m1* Vy1 + m2*Vy2 + m3*Vy3 + m4*Vy4 ) /( m1 +m2+m3+m4)
Vy = (8.3*-3.9 + 8.5*4.9 + 8.6*1.9 + 8.8*-3)/(8.3 + 8.5 + 8.6 + 8.8) = -0.023

Vx of the centre of mass = ( m1* Vx1 + m2*Vx2 + m3*Vx3 + m4*Vx4 ) /( m1 +m2+m3+m4)
Vx = (8.3*3.2 + 8.5*-5.2 + 8.6*-6.2 + 8.8*4.2)/(8.3 + 8.5 + 8.6 + 8.8) = -0.994
therefore speed = root over (Vy^2 + Vx^2)
v = (-0.994^2 + -0.023^2)^1/2
V = 0.994266

when a fifth particle is placed at the origin its x5 = 0, so in the formula m5*x5 = 0.. but its mass is not 0 .
So the value of the denominator increase and the x position of the centre of mass decreases.

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