In exercise physiology studies, it is sometimes important to determine the location of a person's center of gravity. This can be done with the arrangement shown in the figure below. A light plank rests on two scales that read Fg1 = 440 N and Fg2 = 320 N. The scales are separated by a distance of 2.00 m. How far from the woman's feet is her center of gravity?

Question

I have no idea where to start with this question. Any help at all would be appreciated.

Answer 1

To solve this problem, we'll use the principle of torque balance. Torque is a measure of the rotational force acting on an object, and in this case, the torques acting on the plank due to the woman's weight (Fg) and the two scale readings (Fg1 and Fg2) must add up to zero.

First, let's set up our coordinate system: Let's say the left scale (Fg1) is at the origin (x = 0), the right scale (Fg2) is at x = 2.00 m, and the woman's center of gravity (Fg) is at some x value between the two scales (we'll call it x_cg).

Now considering the torques, clockwise torques are considered positive, and counterclockwise torques are considered negative:
1. The torque due to Fg1 is counterclockwise because it tries to rotate the plank counterclockwise around the center of gravity, so its torque is -Fg1 * x_cg.
2. The torque due to Fg2 is clockwise because it tries to rotate the plank clockwise around the center of gravity, so its torque is Fg2 * (2.00 - x_cg).
3. The torque due to the woman's weight Fg is zero because it acts right on the center of gravity.

Now we need to add these torques up and set them equal to zero (since the plank is not rotating):
-Fg1 * x_cg + Fg2 * (2.00 - x_cg) = 0

Now we can plug in the values given in the problem:
-440 * x_cg + 320 * (2.00 - x_cg) = 0

Now we just need to solve for x_cg:
-440 * x_cg + 320 * 2.00 - 320 * x_cg = 0
-440*x_cg + 640 - 320*x_cg = 0
(320 - 440)*x_cg = -640
(-120)*x_cg = -640
x_cg = 640 / 120
x_cg = 5.3333 / 2
x_cg = 2.6667

So the woman's center of gravity is located 2.67 meters from her feet.