A person’s arm is held with the upper arm vertical, the lower arm and hand horizontal. Find the center of mass of the arm in this configuration, given the following data. upper arm has a mass of 2.5kg and a COM 0.18m above the elbow, lower arm has a mass of 1.6kg and a COM 0.12m to the right of the elbow hand has a mass of 0.64kg and a COM 0.40m to the right of the elbow. Holding a 0.8kg water bottle in his hand
98
To find the center of mass of the arm in this configuration, we need to take into account the masses and positions of the upper arm, lower arm, hand, and the water bottle.
First, let's calculate the moments of each component relative to a reference point. We can choose the reference point as the elbow joint.
The moment M is given by the equation: M = m * d, where m is the mass and d is the perpendicular distance from the reference point to the component's center of mass.
For the upper arm:
Mass (m1) = 2.5 kg
Perpendicular distance (d1) = 0.18 m
Moment (M1) = m1 * d1
For the lower arm:
Mass (m2) = 1.6 kg
Perpendicular distance (d2) = 0.12 m
Moment (M2) = m2 * d2
For the hand:
Mass (m3) = 0.64 kg
Perpendicular distance (d3) = 0.4 m
Moment (M3) = m3 * d3
For the water bottle:
Mass (m4) = 0.8 kg
Perpendicular distance (d4) = 0.4 m (assumed to be the same as the hand's COM distance since it is held in the hand)
Moment (M4) = m4 * d4
Now, let's find the total moment of the arm. We add up the individual moments:
Total Moment = M1 + M2 + M3 + M4
Next, we need to find the total mass of the arm. We sum up the masses of the upper arm, lower arm, hand, and water bottle:
Total Mass = m1 + m2 + m3 + m4
Finally, we can calculate the center of mass (CM) using the equation:
CM = Total Moment / Total Mass
By substituting the values into the equations and performing the calculations, we can find the center of mass of the arm in this configuration.