Question

1. You want to calculate the location of the center of gravity of a person lying on his back. His legs have a mass of 20 kg and the center of gravity of his legs is 0.35 m from the bottom of his feet. His trunk and arms have a mass of 50 kg and the center of gravity of these body parts combined is 1.2 m from the bottom of his feet. His head has mass of 5 kg and the center of gravity of the head is 1.6m from the bottom of his feet.

   1. What is the weight of each body segment and the total body weight?

   2. What is the torque produced by each body segment about an axis through the bottom of his feet?

   3. What is the sum of these torques?

   4. How far from the bottom of the feet is his center of gravity?

Answer 1

Gravitational acceleration = g = 9.81 m/s²

Mass of the legs of the person = M₁ = 20 kg

Mass of the trunk and arms of the person = M₂ = 50 kg

Mass of the head of the person = M₂ = 5 kg

Center of gravity of the legs of the person from the bottom of his feet = d₁ = 0.35 m

Center of gravity of the trunk and arms of the person from the bottom of his feet = d₂ = 1.2 m

Center of gravity of the head of the person from the bottom of his feet = d₃ = 1.6 m

Weight of the legs of the person = W₁

W₁ = M₁g

W₁ = (20)(9.81)

W₁ = 196.2 N

Weight of the trunk and arms of the person = W₂

W₂ = M₂g

W₂ = (50)(9.81)

W₂ = 490.5 N

Weight of the head of the person = W₃

W₃ = M₃g

W₃ = (5)(9.81)

W₃ = 49.05 N

Total body weight of the person = W

W = W₁ + W₂ + W₃

W = 196.2 + 490.5 + 49.05

W = 735.75 N

Torque produced by the legs of the person = T₁

T₁ = W₁d₁

T₁ = (196.2)(0.35)

T₁ = 68.67 Nm

Torque produced by the trunk and arms of the person = T₂

T₂ = W₂d₂

T₂ = (490.5)(1.2)

T₂ = 588.6 Nm

Torque produced by the head of the person = T₃

T₃ = W₃d₃

T₃ = (49.05)(1.6)

T₃ = 78.48 Nm

Total torque = T

T = T₁ + T₂ + T₃

T = 68.67 + 588.6 + 78.48

T = 735.75 Nm

Center of gravity of the person from the bottom of his feet = X

X = 1 m

Weight of the legs = 196.2 N

Weight of the trunk and arms = 490.5 N

Weight of the head = 49.05 N

Total weight of the body = 735.75 N

Torque produced by the legs about an axis through the bottom of the person's feet = 68.67 Nm

Torque produced by the trunk and arms about an axis through the bottom of the person's feet = 588.6 Nm

Torque produced by the head about an axis through the bottom of the person's feet = 78.48 Nm

Total torque = 735.75 Nm

Position of center of gravity from the bottom of the person's feet = 1 m