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A rope is wrapped around a pole so that a force of 75 lbs acts on one end and a force of 53 lbs acts on the other end. If the angle between the two forces is 114 degrees what is the resultant force? What angle does the resultant force make with the 53 lbs force?

https://brainmass.com/math/vector-calculus/resultant-force-example-problem-31033

## SOLUTION This solution is FREE courtesy of BrainMass!

Let the direction of the 53 LBS force be the x axis
Angle between the two forces = 114 degrees

x component of the 75 LBS force= 75 Cos 114 = -30.51 LBS
y component of the 75 LBS force= 75 Sin 114 = 68.52 LBS

Total force in the x direction= 22.49 LBS =53-30.51
Total force in the y direction= 68.52 LBS

Resultant force = square root of {(x component)^2 + (y component)^2}= 72.12 LBS

Angle between the resultant and direction of 53 LBS force ( x axis) = tan inverse ( y component / x component)
= tan inverse (68.52/22.49)= 71.82 degrees

Magnitude of resultant force= 72.12 LBS
angle between resultant and 53 LBS force= 71.82 degrees

This can also be solved taking the direction of 75 LBS force as the x axis

Let the direction of the 75 LBS force be the x axis
Angle between the two forces = 114 degrees

x component of the 53 LBS force= 53 Cos 114 = -21.56 LBS
y component of the 53 LBS force= 53 Sin 114 = 48.42 LBS

Total force in the x direction= 53.44 LBS =75-21.56
Total force in the y direction= 48.42 LBS

Resultant force = square root of {(x component)^2 + (y component)^2}= 72.11 LBS

Angle between the resultant and direction of 75 LBS force ( x axis) = tan inverse ( y component / x component)
= tan inverse (48.42/53.44)= 42.18 degrees
Therefore angle between 53 LBS force and resultant = 114-42.18 =71.82