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:

m

x

y

vx

vy

  1  

9 kg

-2.4 m

-4.5 m

2.9 m/s

4.1 m/s

  2  

8.7 kg

-3.7 m

3.4 m

-4.9 m/s

5 m/s

  3  

8.4 kg

4.6 m

-5.4 m

-6.2 m/s

1.9 m/s

  4  

8.1 kg

5.6 m

2.7 m

-4.1 m/s

-2.9 m/s

1) What is the distance the center of mass is from the origin?______m

2) What is the x-component of the velocity of the center of mass?____m/s

3) What is the y-component of the velocity of the center of mass?____m/s

m

x

y

vx

vy

  1  

9 kg

-2.4 m

-4.5 m

2.9 m/s

4.1 m/s

  2  

8.7 kg

-3.7 m

3.4 m

-4.9 m/s

5 m/s

  3  

8.4 kg

4.6 m

-5.4 m

-6.2 m/s

1.9 m/s

  4  

8.1 kg

5.6 m

2.7 m

-4.1 m/s

-2.9 m/s

Explanation / Answer

1) x_cm = (m1 x1 + m2 x2 + m3 x3 + m4 x4)/(m1 + m2 + m3 + m4)

x_cm = [ (9 x -2.4) + (8.7 x -3.7) + (8.4 x 4.6) + (8.1 x 5.6)] / (9 + 8.7 + 8.4 + 8.1)

x_cm = 0.857 m

similarly, y_cm = -1 m

distance = sqrt(x_cm^2 + y_cm^2) = 1.32 m

2) vx_cm = (m1 vx1 + m2 vx2 + m3 vx3 + m4 vx4) / (m1 + m2 + m3 + m4)

= - 2.98 m/s

3) vy_cm = 2.13 m/s

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