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 = 40.5 kg) with a center of mass located on the y axis at a point 0.431 m above the origin, (2) the upper legs (mass = 19.9 kg) with a center of mass located on the x axis at a point 0.172 m to the right of the origin, and (3) the lower legs and feet (total mass = 10.3 kg) with a center of mass located 0.443 to the right of and 0.252 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.

To find the x and y coordinates of the center of mass of the human figure, we can use the concept of weighted averages.

(a) To find the x coordinate of the center of mass:
The x coordinate of the center of mass is given by the equation:
x_cm = (m1 * x1 + m2 * x2 + m3 * x3) / (m1 + m2 + m3)

Given:
m1 = 40.5 kg (mass of torso, neck, and head)
x1 = 0 (x coordinate of the center of mass of torso, neck, and head)
m2 = 19.9 kg (mass of upper legs)
x2 = 0.172 m (x coordinate of the center of mass of upper legs)
m3 = 10.3 kg (mass of lower legs and feet)
x3 = 0.443 m (x coordinate of the center of mass of lower legs and feet)

Substituting the values into the equation:
x_cm = (40.5 * 0 + 19.9 * 0.172 + 10.3 * 0.443) / (40.5 + 19.9 + 10.3)

Calculating the expression:
x_cm = (0 + 3.4308 + 4.5539) / 70.7

x_cm ≈ 0.108 m

Therefore, the x coordinate of the center of mass of the human figure is approximately 0.108 meters to the right of the origin.

(b) To find the y coordinate of the center of mass:
The y coordinate of the center of mass is given by the equation:
y_cm = (m1 * y1 + m2 * y2 + m3 * y3) / (m1 + m2 + m3)

Given:
m1 = 40.5 kg (mass of torso, neck, and head)
y1 = 0.431 m (y coordinate of the center of mass of torso, neck, and head)
m2 = 19.9 kg (mass of upper legs)
y2 = 0 (y coordinate of the center of mass of upper legs)
m3 = 10.3 kg (mass of lower legs and feet)
y3 = -0.252 m (y coordinate of the center of mass of lower legs and feet)

Substituting the values into the equation:
y_cm = (40.5 * 0.431 + 19.9 * 0 + 10.3 * (-0.252)) / (40.5 + 19.9 + 10.3)

Calculating the expression:
y_cm = (17.4555 + 0 - 2.6016) / 70.7

y_cm ≈ 0.177 m

Therefore, the y coordinate of the center of mass of the human figure is approximately 0.177 meters above the origin.