Question

Design an experiment to measure the center of mass of a human. Give equations.

Answer 1

Placing a pivot at the center of mass of a system results in that system being in equilibrium and having a net torque of zero. On each end of a long, rigid body, the torque on one end is equal in magnitude but opposite in direction to the torque on the other end, resulting in a net torque of zero.

Procedure

1-Lay two 2 x 4 wooden beams of approximately 243 cm (8 ft) length side by side across two scales so that the ends of each beam are situated at the center of each scale.

2-Attach two meter sticks end to end on one side of one of the beams. This is done in order to measure one's height when a person is lying down.

3-Record the weight of the beams without anyone on top of them. This is the tare weight.

4-Have a person remove his/her shoes, and then lay across the beams, making sure his/her heels line up with the end of one side of the beams. Have the person lay with their toes pointing upward and hands at their sides.

5-Record the weight on each scale. If the scales are not in the same units, use a conversion to make the data have common units, i.e. kilograms to pounds, pounds to kilograms.

6-Record the height of the person using the meter sticks that are attached to one of the wooden beams.

Analysis

In order to determine the center of mass of a person, we used the formula for torque:

Στ = rF

The torques due to the weight on each end of the person were set equal to each other. The displayed weight on the scale was the force, and there are two different radii. The first radius is the distance to the center of mass from the person's feet, and the second radius is the length of the board minus the distance to the center of mass from the person's feet. The net torque of the system is zero and therefore the torques on the opposite sides of the boards must be equal:

w1x = w2(l-x),

where w1 is equal to the weight at the person's feet, x is equal to the distance from the person's feet to his/her center of mass, w2is equal to the weight at the head, and l is equal to the length of the beams. The resulting formula, when solved for the distance to the center of mass from the person's feet (radius one) is:

x = w2l/(w1 + w2)

After determining the location of each person's center of mass, the ratio of the center of mass to the height of each person was calculated using the formula:

x/h,

where x is the location of the person's center of mass and h is the person's height.