A person is standing on a level floor. His head, upper torso, arms, and hands together weigh 470 N and have a center of gravity that is 1.13 m above the floor. His upper legs weigh 108 N and have a center of gravity that is 0.755 m above the floor. Finally, his lower legs and feet together weigh 87.1 N and have a center of gravity that is 0.250 m above the floor. Relative to the floor, find the location of the center of gravity for the entire body.
total weight = 470+108+87.1
total moment = 470(1.13)+108(.755)+87.1(.25)
cg height = total moment / total weight
To find the location of the center of gravity for the entire body, we need to consider the weight and the vertical position of each segment of the body and then calculate the weighted average of their positions.
First, let's assign variables to the given weights and heights:
Weight of head, upper torso, arms, and hands (W1) = 470 N
Height of center of gravity of head, upper torso, arms, and hands (h1) = 1.13 m
Weight of upper legs (W2) = 108 N
Height of center of gravity of upper legs (h2) = 0.755 m
Weight of lower legs and feet (W3) = 87.1 N
Height of center of gravity of lower legs and feet (h3) = 0.250 m
Now, we can calculate the total weight of the body (W_total) by adding the weights of the different segments:
W_total = W1 + W2 + W3
= 470 N + 108 N + 87.1 N
= 665.1 N
To find the vertical position of the center of gravity of the entire body (h_total), we need to calculate the weighted average of the heights of the different segments:
h_total = (W1 * h1 + W2 * h2 + W3 * h3) / W_total
Substituting the given values:
h_total = (470 N * 1.13 m + 108 N * 0.755 m + 87.1 N * 0.250 m) / 665.1 N
Now, let's calculate this expression:
h_total = (530.1 N*m + 81.54 N*m + 21.775 N*m) / 665.1 N
= 633.445 N*m / 665.1 N
= 0.953 m
Therefore, the center of gravity for the entire body is located 0.953 meters above the floor.