A man who weighs 50.0 N stands 3.00 m from the right end of a board that is supported by two vertical cords attached to either end of the board. The board is 5.00 m long and weighs 10.0 N. What is the tension in each of the two cords?
To find the tension in each of the two cords supporting the board, we need to consider the forces acting on the board and set up an equilibrium equation.
First, let's consider the forces acting on the board. We have the weight of the man (50.0 N) acting downward at a distance of 3.00 m from the right end of the board. We also have the weight of the board itself (10.0 N) acting downward at its center, which is 2.50 m from either end.
There are two cords supporting the board, one at each end. Let's call the tension in the left cord T1 and the tension in the right cord T2.
Now, let's set up the equilibrium equation by considering the forces in the vertical direction. The sum of the vertical forces must be zero for the board to be in equilibrium.
Taking upward forces as positive and downward forces as negative, the equation becomes:
T1 + T2 - (50.0 N + 10.0 N) = 0
Simplifying the equation:
T1 + T2 = 60.0 N
Since we have two unknowns (T1 and T2), we need another equation to solve for both of them. To do this, we can consider the torques acting on the board.
The torque exerted by the man's weight is given by the equation:
Torque = Force x Distance
Torque1 = (50.0 N) x (3.00 m)
The torque exerted by the board's weight is given by the equation:
Torque2 = (10.0 N) x (2.50 m)
In equilibrium, the sum of torques must be zero. Since we have two torques acting in opposite directions, they will have opposite signs.
Torque1 - Torque2 = 0
(50.0 N) x (3.00 m) - (10.0 N) x (2.50 m) = 0
150.0 Nm - 25.0 Nm = 0
125.0 Nm = 0
This equation holds true, which means that the torques are in equilibrium.
Now, let's solve the system of equations to find the tension in each cord:
T1 + T2 = 60.0 N (equation 1)
Torque1 - Torque2 = 0 (equation 2)
From equation 2, we can solve for T1 in terms of T2:
T1 = Torque2 / Distance1
T1 = (10.0 N) x (2.50 m) / (3.00 m)
T1 = 8.33 N
Substituting this value into equation 1, we can solve for T2:
8.33 N + T2 = 60.0 N
T2 = 60.0 N - 8.33 N
T2 = 51.67 N
Therefore, the tension in each of the two cords is approximately 8.33 N and 51.67 N, respectively.