A uniform plank of length 2.00 m and mass 35.0 kg is supported by three ropes, as indicated by the blue vectors in the figure below. Find the tension in each rope when a 690-N person is d = 0.725 m from the left end.
file:///Users/Alex/Desktop/12-p-025.gif
To find the tension in each rope, we need to consider the forces acting on the plank. In this case, the person's weight provides the only external force. The tension in each rope counteracts this force and keeps the plank in equilibrium.
Let's analyze the forces acting on the plank:
1. Weight of the person: The person's weight is given as 690 N. We need to convert this force to the corresponding torque because the person is not positioned at the center of the plank. The torque is given by the formula: torque = force * distance. In this case, the distance is 0.725 m from the left end. Therefore, the torque exerted by the person is torque_person = 690 N * 0.725 m.
2. Tension in the left rope: Denote this tension as T1. The T1 rope pulls upward at a distance of 0.725 m from the left end of the plank. Therefore, the torque exerted by T1 is torque_T1 = T1 * 0.725 m.
3. Tension in the right rope: Denote this tension as T2. The T2 rope pulls upward at a distance of (2.00 - 0.725) m from the left end of the plank. Therefore, the torque exerted by T2 is torque_T2 = T2 * (2.00 - 0.725) m.
4. Tension in the middle rope: Denote this tension as T3. The T3 rope pulls upward at a distance of 1.00 m from the left end of the plank. Therefore, the torque exerted by T3 is torque_T3 = T3 * 1.00 m.
Since the plank is in equilibrium, the sum of torques acting on it must be zero: torque_person + torque_T1 + torque_T2 + torque_T3 = 0.
Now we can set up the equation and solve for the tensions:
torque_person + torque_T1 + torque_T2 + torque_T3 = 0
(690 N * 0.725 m) + (T1 * 0.725 m) + (T2 * 1.275 m) + (T3 * 1.00 m) = 0
We have three unknowns (T1, T2, T3) and one equation. However, we can use the fact that the sum of tensions equals the person's weight to set up a second equation:
T1 + T2 + T3 = 690 N
Now, we have a system of two equations and three unknowns. We can solve this system of equations simultaneously using mathematical techniques like substitution or elimination to find the values of T1, T2, and T3.