The three forces (magnitudes 100 lb,200 lb, and P) combine to produce a resultan

Question

The three forces (magnitudes 100 lb,200 lb, and P) combine to produce a resultant R (Figure Below). The three forces act in known directions, but the numerical value P is unknown.

(a)What should the magnitude of P be so that the resultant is as small as possible?

Explanation / Answer

Good Problem First you need an equation for the resultant. In general the magnitude of a combination of vectors uses the Pythagorean theorem ==> R = sqrt( (sum of x's)^2+(sum of y's)^2) Summing all the x components: Sx = 200 cos(90) + 100 cos(60) + p cos(-40) = 50 + .766p Summing all the y components: Sy = 200 sin(90) + 100 sin(60) + p sin(-40) = 286.6 - 0.643p So R(p) = sqrt[ (50+.766p)^2 + (286.6 - .643p)^2] Take the derivative of this expression and set it equal to zero...then solve for p This will give you p = 146 lb To find the angle, add all of your vector components, which you can since you now know p, and use trig: tan ? = Sy/Sx ? = atan(Sy/Sx)

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