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.
(470*1.13 + 108*0.755 + 87.1*0.250)/(470+108+87.1) = 0.954
The figure shows a person whose weight is W = 500 N doing push-ups. Find the normal force exerted by the floor on (a) each hand and (b) each foot, assuming that the person holds this position.
To find the location of the center of gravity for the entire body, we can use the principle of moments. The total moment of the body can be found by summing the individual moments of each segment.
The moment of an object is calculated by multiplying its weight by the distance from the reference point (in this case, the floor). The total moment is then divided by the total weight of the body to find the center of gravity.
Let's calculate the total moment and weight of the body:
Moment of the head, upper torso, arms, and hands:
Moment = 470 N * 1.13 m = 531.1 Nm
Moment of the upper legs:
Moment = 108 N * 0.755 m = 81.84 Nm
Moment of the lower legs and feet:
Moment = 87.1 N * 0.250 m = 21.775 Nm
Total moment = 531.1 Nm + 81.84 Nm + 21.775 Nm = 634.715 Nm
Total weight = 470 N + 108 N + 87.1 N = 665.1 N
Now, let's find the location of the center of gravity:
Center of gravity = Total moment / Total weight
Center of gravity = 634.715 Nm / 665.1 N = 0.954 m above the floor
Therefore, the center of gravity for the entire body is located 0.954 m above the floor.
To find the location of the center of gravity for the entire body, we need to calculate the overall weight and the vertical position of the center of gravity.
First, let's calculate the total weight:
Weight of the head, upper torso, arms, and hands = 470 N
Weight of the upper legs = 108 N
Weight of the lower legs and feet = 87.1 N
Total weight = 470 N + 108 N + 87.1 N = 665.1 N
Next, we need to calculate the weighted average of the vertical positions of the center of gravity.
Weight of the head, upper torso, arms, and hands = 470 N
Vertical position (height) of the center of gravity for the head, upper torso, arms, and hands = 1.13 m
Weighted contribution = Weight × Vertical position
= 470 N × 1.13 m = 530.1 N·m
Weight of the upper legs = 108 N
Vertical position (height) of the center of gravity for the upper legs = 0.755 m
Weighted contribution = Weight × Vertical position
= 108 N × 0.755 m = 81.64 N·m
Weight of the lower legs and feet = 87.1 N
Vertical position (height) of the center of gravity for the lower legs and feet = 0.250 m
Weighted contribution = Weight × Vertical position
= 87.1 N × 0.250 m = 21.775 N·m
Now, we can find the total weighted vertical position of the center of gravity:
Total weighted vertical position = Sum of the weighted contributions / Total weight
= (530.1 N·m + 81.64 N·m + 21.775 N·m) / 665.1 N
= 633.535 N·m / 665.1 N
= 0.952 m
Therefore, the location of the center of gravity for the entire body, relative to the floor, is 0.952 m above the floor.