A woman who weighs 500N stands on an 8.0-m-long board that weighs 100N. The board is supported at each end. The support force at the right end is 3 times the support force at the left end. How far from the right end is the woman standing?
1.6
To solve this problem, we can start by using the principle of moments. The principle of moments states that the sum of the moments acting on an object in equilibrium is equal to zero.
Let's take moments about the left end of the board. The weight of the board (100N) can be considered as acting at its midpoint, which is 4.0m from the left end. Therefore, the moment created by the weight of the board is 100N * 4.0m = 400Nm.
Now let's consider the woman's weight. The woman weighs 500N and is standing at a distance from the left end of the board, which we can call x.
The support force at the left end can be represented as a reaction force, R1, acting upwards. The support force at the right end can be represented as a reaction force, R2, acting upwards.
The moment created by the woman's weight is 500N * x. Since we have already taken moments about the left end of the board, this moment acts in the opposite direction compared to the board's weight moment. Therefore, we can write this moment as -500N * x.
According to the principle of moments, the sum of the moments acting on the board should be equal to zero:
400Nm - 500N * x = 0
Simplifying the equation, we get:
500N * x = 400Nm
Dividing both sides of the equation by 500N, we can solve for x:
x = 400Nm / 500N
x ≈ 0.8m
Therefore, the woman is standing approximately 0.8m from the right end of the board.
To solve this problem, we need to consider the forces acting on the board.
Let's denote the support force at the left end of the board as FL and the support force at the right end as FR.
The total weight of the woman and the board is 500N + 100N = 600N, which is acting downward. Since the board is in equilibrium (not moving), the total upward force acting on it must be equal to the total downward force.
We can set up the equation:
FL + FR = 600N
Now we know that the support force at the right end is 3 times the support force at the left end:
FR = 3FL
Substituting this equation into the previous equation, we have:
FL + 3FL = 600N
Combining like terms, we get:
4FL = 600N
Dividing both sides by 4, we find:
FL = 150N
Now that we know the support force at the left end, we can find the support force at the right end:
FR = 3FL = 3 * 150N = 450N
The woman's weight (500N) is acting downward at the right end of the board. So, to calculate the distance she is standing from the right end, we can consider the torques (moments) acting on the board.
The torque due to the woman's weight (500N) is given by the equation:
Torque = Force x Distance
If the woman is standing at a distance x from the right end, the torque due to her weight is:
Torque = 500N * x
The torque due to the support force at the left end (FL) is zero since it acts at a distance of 0 meters from the left end.
The torque due to the support force at the right end (FR) is given by:
Torque = FR * (8m - x)
Since the board is in equilibrium, the total torque must be zero:
500N * x + FR * (8m - x) = 0
Substituting the known values, we have:
500N * x + 450N * (8m - x) = 0
Expanding and simplifying the equation, we find:
500N * x + 450N * 8m - 450N * x = 0
50N * x = 450N * 8m
x = (450N * 8m) / 50N
x = 3600N-m / 50N
x = 72m
Therefore, the woman is standing 72 meters from the right end of the board.
Let the left end support force be F1 and the right end support force = F2
3F1 = F2 (You are told this)
F1 + F2 = (4/3)F2 = 600 N (From vertical force balance)
F2 = 450 N ; F1 = 150 N
Let the woman's distance from left end = x.
Apply a moment balance. You can apply it anywhere, but chose the left end.
500 x +100*4 = F2*8 = 3600
Complete the solution for x. It will be in meters