Question

A tabletop gamer has designed a game that requires three dice to be thrown onto a tray with a measurement grid. To add an extra degree of randomness, the coordinates of the center of mass of the three dice are used as well. The masses of the three dice are 12.30 g, 13.70 g, and 19.90 g, and their respective coordinates after one particular throw are (0.2270 m,-0.3270m) and (-0.3370m,0.4150m) and (-0.3630, -0.3170m)

What are the resulting coordinates of the center of mass of the dice x and y ?

Answer 1

m₁ = 12.30 g

x₁ = 0.2270 m

y₁ = - 0.3270 m

m₂ = 13.70 g

x₂ = - 0.3370 m

y₂ = 0.4150 m

m₃ = 19.90 g

x₂ = - 0.3630 m

y₂ = - 0.3170 m

x-coordinate of the center of mass is given as

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

x_cm = ((12.30) (0.2270) + (13.70) (- 0.3370) + (19.90) (- 0.3630))/(12.30 + 13.70 + 19.90)

x_cm = - 0.1971 m

y-coordinate of the center of mass is given as

y_cm = (m₁ y₁ + m₂ y₂ + m₃ y₃)/(m₁ + m₂ + m₃)

y_cm = ((12.30) (- 0.3270) + (13.70) (0.4150) + (19.90) (- 0.3170))/(12.30 + 13.70 + 19.90)

y_cm = - 0.1012 m