What is the tension in the cable?

Question

A sign with a mass of 1000.0 kg is suspended from a wall witha cable that is attched to the wall at a point 3 m above a horizontal beam that causes the sign to be a distance of 4 m from the wall.
What is the tension in the cable?

Answer 1

The sign is placed at the far end of the beam so that it is 4 meters from the wall. At the same spot, the cable is connected to the beam, which extends from the far side of the beam to the wall.

 

The entire contraption looks like a triangle, more or less. If you draw a force diagram (free body diagram) you should have the weight of the sign pointing down at the end of the beam and the tension in the cable pointing up and to the side.

 

What is the angle between the beam and the tension? We can figure that out using trig. theta = inverse tan (3/4) = 36.87 degrees.

 

Now we need to do a sum of torques. Recall that a torque is a force * distance if they are perpendicular. Another way to think about it is F*d*sin(theta), where theta is the angle between the force and distance. 

 

The torque caused by the weight of the sign and the torque caused by the tension are in the opposite directions. This should be fairly clear from the sketch, but if it's not, please respond and ask if you have a question about that.

 

So the torque caused by the weight of the sign is

 

mass * grav accel * distance from wall. The distance and force are 90 degrees apart, so sin(90) = 1, and we are left with

 

1000.0 * 9.8 * 4 = 39200 Nm

 

Now we look at the torque caused by the tension. We don't know the tension force (which is what we want to find), but it makes an angle with the distance form the wall. That angle is what we found before...36.87 deg

 

We now use the torque formula again

 

T * 4 * sin (36.87)

 

Since these torques point in opposite directions, and the whole thing isn't moving, they have to be equal to each other. This is similar to the idea that the sum of forces is =0 when an object isn't moving.

 

39,200 = T * 4 * sin(36.87)

 

Solve for T and we get

 

16,300 N