A 1210 N uniform beam is attached to a vertical wall at one end and is supported by a cable at the other end. A W = 1820 N crate hangs from the far end of the beam. The angle between the beam and the horizontal is 30 degrees upwards and the angle between the horizontal and the cable attached to the wall is 50 degrees upward.

Calculate the magnitude of the tension in the wire.

Beam length a

t = tension in cable

moments clockwise at wall hinge
1210(a/2)cos 30 + 1820 a cos 30 - t a sin 50 cos 30 = 0 (note a cancels, is irrelevant)

524 + 1576 =.663 t
solve this for t

To calculate the magnitude of the tension in the wire, we need to consider the equilibrium of forces acting on the beam. We'll break down the forces into their horizontal and vertical components.

Let's start by drawing a diagram:

```
|\
| \
|__\_______

\ / W
\ /
\ /
\ /
\
\
\

```

In this diagram:
- The vertical wall is on the left side.
- The cable providing support to the beam is on the right side, attached to the beam perpendicularly.
- The weight of the crate W is acting downward.

Now, let's analyze the forces acting on the beam:

1. Weight of the beam:
The weight of the beam acts downward at the center of the beam. Its magnitude is given as 1210 N.

2. Reaction force at the wall:
Since the beam is attached to a vertical wall, there will be a reaction force acting horizontally from the wall to the beam. Let's denote this force as R. This reaction force only has a horizontal component since the beam is smooth and there is no friction involved.

3. Tension in the wire:
The tension in the wire is acting upward, opposite to the weight of the crate.

To find the magnitude of the tension in the wire, we need to equate the vertical and horizontal components of forces to zero, as the beam is in equilibrium.

Vertical equilibrium:
Sum of upward forces = Sum of downward forces

In this case, the only vertical force acting is the vertical component of the tension in the wire.

So, we can write:
Tsin(30°) = W
Tsin(30°) = 1820 N

Now, we can solve this equation to find the value of T (the tension in the wire).

T = 1820 N / sin(30°)

Calculating this using a scientific calculator gives us:

T ≈ 3640 N

Therefore, the magnitude of the tension in the wire is approximately 3640 N.