Question

The drawing shows a human figure approximately in a sitting position. For the purposes of this problem, there are three parts to the figure, and the center of mass of each one is shown in the drawing. These parts are: (1) the torso, neck, and head (total mass = 43.4 kg) with a center of mass located on the y axis at a point 0.374 m above the origin, (2) the upper legs (mass = 16.3 kg) with a center of mass located on the x axis at a point 0.165 m to the right of the origin, and (3) the lower legs and feet (total mass = 10.6 kg) with a center of mass located 0.414 to the right of and 0.271 m below the origin. Find the (a) x coordinate and (b) the y coordinate of the center of mass of the human figure. Note that the mass of the arms and hands (approximately 12% of the whole-body mass) has been ignored to simplify the drawing.

Answer 1

For a system of bodies and you get the position vector of center of mass by
R = Σ (m_i · r_i ) / Σ m_i
where m_i is the mass of a body and r_i the position vector of its center of mass.

If expand the vector equation to its components you get:
x = Σ (m_i · x_i ) / Σ m_i
y = Σ (m_i · y_i ) / Σ m_i


(a)

x = (m1*x1 + m2*x2 + m3*x3 ) / (m1 + m2 + m3)
x = (43.4kg*0m + 16.3kg *0.165m + 10.6kg*0.414m ) / (43.4 kg + 16.3 kg + 10.6 kg)
x = 7.07 / 70.3 kg = 0.100 m

(b)

y = (m1*y1 + m2*y2 + m3*y3 ) / (m1 + m2 + m3)
y = (43.4kg*0.374m + 16.3kg *0m + 10.6kg*(-0.271 m) ) / (43.4 kg + 16.3 kg + 10.6 kg)
y = 13.359 / 70.3

y = 0.190 m