This is one way to measure a person's center of mass. A person lies on a plank on two scales: one at her head and the other at her feet. The scales are 1.69 meters apart. Her head scale reads 400 N, and her foot scale reads 322 N. How far from her feet is her center of mass?

To find the center of mass, we need to consider the concept of torque. Torque is the force applied at a distance from a pivot point, and it causes an object to rotate. In this case, the pivot point is the center of mass.

To determine the distance of the person's center of mass from her feet, we can use the equation:

Torque = Force × Distance

Let's calculate the torques at the head and feet scales. The distance between the scales is given as 1.69 meters. The force at the head scale is 400 N, and at the foot scale, it is 322 N.

At the head scale, the torque is:

Torque_head = Force_head × Distance

Torque_head = 400 N × 1.69 m

Torque_head = 676 N·m

Similarly, at the foot scale, the torque is:

Torque_foot = Force_foot × Distance

Torque_foot = 322 N × 1.69 m

Torque_foot = 544.18 N·m

To find the distance of the center of mass from the feet, we can equate the torques at the head and feet scales:

Torque_head = Torque_foot

Force_head × Distance_head = Force_foot × Distance_foot

400 N × Distance_head = 322 N × Distance_foot

Distance_foot = (400 N × Distance_head) / 322 N

Plugging in the values:

Distance_foot = (400 N × 1.69 m) / 322 N

Distance_foot ≈ 2.1 m

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