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. We will calculate the total moments about the x and y axes and divide them by the total mass of the figure.
Let's start with the x coordinate of the center of mass (a).
The x coordinate is given by the weighted average of the individual x coordinates of the three parts, where the weight is the mass of each part.
The x coordinate of the torso (part 1) is 0 because it lies on the y-axis.
The x coordinate of the upper legs (part 2) is 0.172 m to the right of the origin.
The x coordinate of the lower legs and feet (part 3) is 0.443 m to the right of the origin.
The total mass of the figure is the sum of the masses of the three parts: 40.5 kg (torso) + 19.9 kg (upper legs) + 10.3 kg (lower legs and feet) = 70.7 kg.
Now, we can calculate the x coordinate of the center of mass:
(a) x coordinate = (mass of part 1 * x coordinate of part 1 + mass of part 2 * x coordinate of part 2 + mass of part 3 * x coordinate of part 3) / total mass
(a) x coordinate = (40.5 kg * 0 + 19.9 kg * 0.172 m + 10.3 kg * 0.443 m) / 70.7 kg
(a) x coordinate = (3.428 kg*m + 3.589 kg*m) / 70.7 kg
(a) x coordinate = 7.017 kg*m / 70.7 kg
(a) x coordinate = 0.099 m
Therefore, the x coordinate of the center of mass of the human figure is approximately 0.099 m to the right of the origin.
Now, let's find the y coordinate of the center of mass (b).
The y coordinate is given by the weighted average of the individual y coordinates of the three parts, where the weight is the mass of each part.
The y coordinate of the torso (part 1) is 0.431 m above the origin.
The y coordinate of the upper legs (part 2) is 0 because they lie on the x-axis.
The y coordinate of the lower legs and feet (part 3) is 0.252 m below the origin.
Using the same total mass of 70.7 kg, we can calculate the y coordinate of the center of mass:
(b) y coordinate = (mass of part 1 * y coordinate of part 1 + mass of part 2 * y coordinate of part 2 + mass of part 3 * y coordinate of part 3) / total mass
(b) y coordinate = (40.5 kg * 0.431 m + 10.3 kg * (-0.252 m)) / 70.7 kg
(b) y coordinate = (17.445 kg*m - 2.6076 kg*m) / 70.7 kg
(b) y coordinate = 14.8374 kg*m / 70.7 kg
(b) y coordinate = 0.210 m
Therefore, the y coordinate of the center of mass of the human figure is approximately 0.210 m above the origin.
In summary:
(a) The x coordinate of the center of mass is approximately 0.099 m to the right of the origin.
(b) The y coordinate of the center of mass is approximately 0.210 m above the origin.