Question

A light plank rests on two scales that read Fg1 = 410 N and Fg2 = 310 N. The scales are separated by a distance of 2.00 m. How far from the woman's feet is her center of gravity (the woman is laying on a plank with scales on both ends supporting the woman and plank)?





You have three unknowns: weight of woman, weight of plank, and cg for the woman.
Assume the womans feet is at Fg1.
wplk is weight of plank
woman is weight of woman
d is the distance from her feet.

Writing equations:
1) about Fg1
310*2 - wplk*2 - woman*d=0
2) about fg2:
410*2 - wp1k*2 - woman(2-d)=0
3) about the center (cg of board)
310*1 + woman*(d-1) -410*1=0

Three equations, three unknowns.

Is it possible to solve for the three unknowns with the information given?

Three independent equations says you can. I had the same thought when I started. See if those equations lead to a solution for it. In my mind, d depends on the weight of the woman vs plank. So it may not lead to a solution.

Answer 1

We can simplify the system of equations as follows:

1) 620 - 2*wplk - woman*d=0 -> 2*wplk + woman*d = 620
2) 820 - 2*wplk - 2*woman + woman*d=0 -> 2*wplk - woman*(2-d) = 820
3) 310 + woman*(d-1) - 410=0 -> woman*(d-1) = 100 -> woman*(d-1) = 100

From equation 1 and 2, we can conclude that:
woman*d = 620 - 2*wplk
woman*(2 - d) = 820 - 2*wplk

Now, we can solve for d:
woman*d = woman*(2-d) + 2**woman
d = 2 - d + 2*(820 - 2*wplk)/woman
2*d = 2 + 1640/woman - 4*wplk/woman
d = 1 + 820/woman - 2*wplk/woman

Lastly, let's substitute the third equation into the simplified equation obtained above:
woman*(d-1) = 100
woman*(820/woman - 2*wplk/woman)=100

Now, we can cancel out woman from both sides:
820 - 2*wplk = 100
2*wplk = 720

So, from equation 1:
woman*d = 620 - 720
woman*d = -100

But, this is an invalid equation, as weight, and distance cannot be negative. Therefore, the given information is not sufficient to find the distance d from the woman's center of gravity to her feet, unless further assumptions are made.