Tugboats and Tension?

Question

Theres two tug boats, the magnitude of the resultant vector is 6000lbs. And each towline makes an angle of theta for both lines along a single axis.

What is the tension, if theta = 20 degrees?
And what is the Tension equation in respect of theta being a variable?
I guess if i know the equation i can find the 20degrees.

Answer 1

Presumably the two tows are pulling something with a resistive force of 6,000 lbs; so that f = ma = 0 = (P - F); where P is that 6,000 lbs pull, m is the total mass of that something and the two tows, and a = 0 is the acceleration of that total mass.

The resisting force F will not change, no matter what theta is; so we can set F = P = 2T cos(theta); where T is the tension in each line and the 2 reflects there are two tows. Thus T = P/(2 cos(theta)); you can do the math.

As a check, set theta = 0, which means the tows are on top of each other straight in front of the thing they are towing. In which case P = 2T or the pull force is just twice the tension on each line. And that is an intuitively pleasing result. Then, as a further check, set theta = 90 degrees and see what you get. Does that result make sense?

Answer 2

Sounds like the two tugboats each tow equally on something? Although I'm somewhat confused by the wording.

If this is the case, then they both two with 3000lbs, opposite the resultant vector and 20 degrees off of their path.

thus

T*sin(theta)=3000

or

T=3000/sin(20)