find the resultant pull on the vessel and the angle bet the ropes and the side of the canal.

Question

Two locomotives on opposite banks of a canal pull a vessel moving parallel to the banks by means of two horizontal ropes. The tensions in the ropes are 200lb and 2400lb respectively, while the angle between them is 60degrees. find the resultant pull on the vessel and the angle bet the ropes and the side of the canal.

Answer 1

The triangle or parallelogram of forces follows the Cosine Rule. The diagram below is a representation of the problem. The blue lines represent the banks of the canal and the solid red lines represent the ropes and the tension in them. W is the vessel. WX has a tension of 200lbs and WZ a tension of 2400lbs. The green line WY is the resultant force and also the direction in which the vessel moves as it is being towed. ∠XWZ=60°, the angle between the ropes. Therefore ∠YZW=120°. WY=√(WZ²+YZ²-2WZ.YZcos120°). Length YZ=WX (parallel sides of parallelogram WZYZ).

WY=√(2400²+200²+2(2400)(200)/2)=√(5800000+480000)=√6280000=2506lbs approx. The angle YWZ has the same measure as the the angle between rope WZ and the bank, and ∠XYW has the same measure as ∠YWZ. XY=WZ=2400.

(Note that the solid red lines do not necessarily represent the lengths of the ropes, which will depend on the type of material, strength, and other factors.)

Drop a perpendicular XN from X on to WZ, then XN=WXsin60° and WN=WXcos60°. 

tanXYW=XN/(XY+WN)=200sin60°/(2400+200cos60°)=100√3/(2400+100)=√3/25. So ∠XYW=tan^-1(√3/25)=3.96° approx.

To find the angle of the other rope to the bank we can see that this angle=180°-(∠YZW+∠XYW)=180-(120+3.96)=56.04° approx. Note that 56.04°+3.96°=60°, from ∠XWZ=∠XWY+∠YWZ, so we could have used this fact to find the required angles that the ropes make with the banks.