A heavy piece of equipment(85kg) is being moved by elevator between floors in a building. Two identical small cranes are used. A cable connecting each crane to a mounting hook on the top of the piece of equipment.The cable from one crane is at an angle of 22 degree below the horizontel while tthe cable from the other hangs at 40 degree. find the tension in each cable if the elevator is accelerating upwords at a constant 0.2ms-2 and include a vector diagram.

Question

A heavy piece of equipment(85kg) is being moved by elevator between floors in a building. Two identical small cranes are used. A cable connecting each crane to a mounting hook on the top of the piece of equipment.The cable from one crane is at an angle of 22 degree below the horizontel while tthe cable from the other hangs at 40 degree. find the tension in each cable if the elevator is accelerating upwords at a constant 0.2ms-2 and include a vector diagram.

Answer 1

To find the tension in each cable, we can break down the forces acting on the piece of equipment and use Newton's second law. Let's start by drawing a vector diagram.

Here is the vector diagram:
```

θ2 = 40°
───────────────■
│ ★ │
│ │
│ │
│ F2 │
│ │
│ │
────┴───────■──────┴──────────
│
│ Fg
│
★
θ1= 22°
```
In the diagram, F1 and F2 represent the tension forces in each cable. Fg represents the gravitational force acting on the equipment, and it is equal to the weight of the equipment, which is given as 85 kg.

To calculate the tension forces, we need to resolve the forces along the vertical and horizontal directions.

Resolving forces along the vertical direction:
For the first cable with an angle of 22 degrees, the vertical component of tension is:
F1_vertical = F1 * sin(22°)

For the second cable with an angle of 40 degrees, the vertical component of tension is:
F2_vertical = F2 * sin(40°)

Resolving forces along the horizontal direction:
For the first cable with an angle of 22 degrees, the horizontal component of tension is:
F1_horizontal = F1 * cos(22°)

For the second cable with an angle of 40 degrees, the horizontal component of tension is:
F2_horizontal = F2 * cos(40°)

Now, let's write the equations. The sum of the vertical forces should be equal to the gravitational force:
F1_vertical + F2_vertical - Fg = 0

Then, the sum of the horizontal forces should be equal to the mass of the equipment times its acceleration:
F1_horizontal + F2_horizontal = mass * acceleration

Since the mass is given as 85 kg and the acceleration is given as 0.2 m/s^2, we can substitute these values into the equation.

Next, using trigonometric functions, we can rewrite the equations as:
F1 * sin(22°) + F2 * sin(40°) - weight = 0
F1 * cos(22°) + F2 * cos(40°) = mass * acceleration

Now, we have a system of two equations with two unknowns (F1 and F2). We can solve these equations simultaneously to find the tension in each cable.