Tension Force

Question

Consider a long 15m uniform wooden beam (mass=60kg) attached horizontally to a wall that can only support a vertical load. The horizontal component of this force of the wall on the beam is identically zero. There is a chandelier of 40kg hanging at a distance of 4.2m from the end of the beam that is attached to the wall. There is a vertical cable hanging down from the ceiling that is attached to the other end of the beam that is in mid air. The entire structure is in static equalibrium. Determine the tension in the vertical cable. 

The tension is one of the following: 404N; 613N; 157N; 890N

 

I spent hours on this and cannot arrive at one of the answers.

 

thank you

Answer 1

I got the tension = 404 N. I made the junction of the wall and the beam to be the pivot point and solved it as a torque problem. If the wall is the pivot point, there are then three forces (and torques) acting on the beam.

 

Torque = F x lever arm

 

THE FORCES

Two are clockwise (downward)

-the weight of the chandelier

- the weight of the board itself

 

One is counterclockwise (upward)

 

-the tension in the cable at the other end of the beam

 

The torque of the tension in the cable is what holds the beam up horizontally- it must be equal and opposite to the combined torques of the chandelier and the board itself.

 

 

The force of the board itself is its weight, mg, or 60 kg x 9.8 m/s²  .   We choose the lever arm length to be half the length of the beam, 7.5 m. By convention, the weight of the object itself is generally assumed to be acting on the center of the object.

 

The force due to the chandelier is its weight, mg, 40 kg x 9.8 m/s². Its lever arm length is 4.2 m, as given to us in the passage.

 

We do not know the tension force- that's what we're looking for. However, we do know the lever arm for the tension. The cable is attached at the end of the beam, the beam is 15 m long. Therefore, the lever arm of the tension torque is 15 m.

 

 

Now we have, in static equilibrium, that the net torque is equal to zero.

That means the sum of the clockwise torque= sum of the counterclockwise torque.

 

Torque of chandelier + Torque of beam itself = Torque of tension in the cable.

(392 N x 4.2 m) + (588 N x 7.5 m) = F tension x 15 m

 

solving for the unknown, F tension, you get 404 N.

 

 

I hope this helps!