A 500N person stands 2.5m from a wall against which a horizontal beam is attached. The beam is 6m long and weighs 200N. A cable is attached to the free end of the beam and makes an angle of 45 to the horizontal and is attached to the wall.

1)Determine the magnitude of the tension of the cable.
2)Determine the reaction force that the wall exerts on the beam.

Jend

To solve the problem, we need to analyze the forces acting on the beam. Let's break it down step by step.

Step 1: Determine the weight of the beam.
Given that the weight of the beam is 200N.

Step 2: Determine the vertical component of the tension in the cable.
The vertical component of the tension in the cable counteracts the weight of the beam and the person standing on it. Since the person's weight is 500N, and the weight of the beam is 200N, the total weight being supported by the cable is 500N + 200N = 700N.

The vertical component of the tension in the cable can be found using trigonometry:
Vertical component of tension = Tension in cable * sin(angle with horizontal)
Vertical component of tension = Tension in cable * sin(45°)

Step 3: Determine the horizontal component of the tension in the cable.
The horizontal component of the tension in the cable opposes the horizontal reaction force exerted by the wall on the beam.

The horizontal component of the tension in the cable can also be found using trigonometry:
Horizontal component of tension = Tension in cable * cos(angle with horizontal)
Horizontal component of tension = Tension in cable * cos(45°)

Step 4: Determine the tension in the cable.
Solving Steps 2 and 3 simultaneously, we can set up the following equations:

Vertical component of tension = 700N
Horizontal component of tension = Reaction force from wall

Using the Pythagorean theorem, we can find the magnitude of the tension in the cable:
(Tension in cable)^2 = (Vertical component of tension)^2 + (Horizontal component of tension)^2

Now, we can solve the equations to find the answers to each question.

1) Determine the magnitude of the tension of the cable:

Substituting the values into the equation from Step 4,
(Tension in cable)^2 = (700N)^2 + (Reaction force from wall)^2

Solving for the tension in the cable, we get:
Tension in cable = sqrt((700N)^2 + (Reaction force from wall)^2)

2) Determine the reaction force that the wall exerts on the beam:

Since the beam is in equilibrium, the sum of the vertical forces acting on it must be zero. This means that the vertical component of the tension in the cable must equal the total weight being supported:
Vertical component of tension = 700N

The reaction force from the wall is equal to the horizontal component of the tension in the cable:
Reaction force from wall = Tension in cable * cos(45°)

Now, substitute the values obtained in step 1 into this equation to find the reaction force from the wall.

Note: Without specific information about the angle between the beam and the wall or any other additional forces, this calculation assumes a simple equilibrium scenario.

To determine the answers to these questions, we need to apply the principles of equilibrium. In this situation, the beam is in equilibrium when the sum of all the forces acting on it is equal to zero.

1) Magnitude of the tension in the cable:
The cable's tension can be resolved into its horizontal and vertical components. Since the cable makes an angle of 45° with the horizontal, both components will have the same magnitude.
Let's denote the magnitude of the tension by T.

Vertical component: T * sin(45°) = T/sqrt(2)
Horizontal component: T * cos(45°) = T/sqrt(2)

For the vertical equilibrium of the beam:
The weight of the person acts downward and is balanced by the vertical component of the tension in the cable:
500N = T/sqrt(2)

Solving for T gives:
T = 500N * sqrt(2)

Therefore, the magnitude of the tension in the cable is 500N * sqrt(2), which is approximately 707N.

2) Reaction force exerted by the wall on the beam:
For horizontal equilibrium, the horizontal forces acting on the beam must balance out.
Let's denote the reaction force exerted by the wall on the beam as R.

Horizontal forces:
The horizontal component of the tension in the cable acts to the left, while the weight of the beam acts to the right.
R = T/sqrt(2) + 200N

Substituting the value of T we found previously:
R = (500N * sqrt(2))/sqrt(2) + 200N

Simplifying the expression:
R = 500N + 200N

Therefore, the reaction force exerted by the wall on the beam is 700N.