Question

Sam, Choi, Mich, Pia, and Sheen are on a 6’ x 4’ rectangular balsa. They are having trouble keeping the balsa balanced as they are unaware of their center of mass. Find their center of mass, given the following:

Sam is at 3 feet east, and 2 feet north of the balsa’s geometric center. She weighs 78 kg.

Choi is 2 feet east, and 2 feet south of the geometric center. He weighs 52 kg.

Mich is the geometric center of the balsa. He weighs 45 kg.

Pia is at 4 feet west, and 2 feet south of the geometric center. She weighs 56 kg.

Sheen is at 1 foot west of the balsa’s geometric center. She weighs 75 kg.

Answer 1

Finally, we have all info to find the center of mass! Choose the system of coordinates: the geometric center is zero, up/right is positive, left/down is negative:

$$y_c=\frac{y_1m_1+y_2m_2+y_3m_3+y_4m_4+y_5m_5}{m_1+m_2+m_3+m_4+m_5},\\\space\\ y_c=\frac{2·78+(-2)·52+0·45+(-2)·56+0·75}{78+52+45+56+75}=0.2\text{ ft South}.\\\space\\ x_c=\frac{x_1m_1+x_2m_2+x_3m_3+x_4m_4+x_5m_5}{m_1+m_2+m_3+m_4+m_5},\\\space\\ x_c=\frac{3·78+2·52+0·45+(-4)·56+(-1)·75}{78+52+45+56+75}=0.13\text{ ft East}.$$