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.
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.
Yes, it is possible to solve for the three unknowns (weight of the woman, weight of the plank, and the position of the woman's center of gravity) with the given information. However, the equations you provided may not lead to a unique solution for the unknowns. The position of the woman's center of gravity (d) depends on the weight distribution between the woman and the plank. Therefore, the weights of the woman and the plank need to be known or assumed in order to solve for d.
Yes, it is possible to solve for the three unknowns - the weight of the woman, the weight of the plank, and the distance from the woman's feet to her center of gravity.
Let's go through the equations one by one:
1) Start by taking the sum of the torques about Fg1 (the left scale):
310 * 2 - wplk * 2 - woman * d = 0
This equation represents the torques caused by the weight on the right scale (310 N) and the weight of the plank (wplk) about Fg1, balanced by the torque caused by the woman's weight (woman) about her center of gravity (d). Since these torques must balance out, the equation is set equal to zero.
2) Next, take the sum of the torques about Fg2 (the right scale):
410 * 2 - wplk * 2 - woman * (2 - d) = 0
Similar to the previous equation, this equation represents the torques caused by the weight on the left scale (410 N) and the weight of the plank (wplk) about Fg2, balanced by the torque caused by the woman's weight (woman) about her center of gravity (2 - d).
3) Finally, take the sum of the torques about the center of gravity (cg) of the plank:
310 * 1 + woman * (d - 1) - 410 * 1 = 0
This equation represents the torques caused by the weight on the left scale (310 N) and the woman's weight (woman) about the center of gravity of the plank (1), balanced by the torque caused by the weight on the right scale (410 N).
Now, you have three equations with three unknowns: wplk, woman, and d. Solve these equations simultaneously to find the values of these unknowns.