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The sum of the magnitudes of two forces acting at a point is 18 and the magnitude of the resultant is 12. If the resultant is at 90o with the force of smaller magnitude, what are the magnitudes of the two forces?

10 years ago

Dear Dennis,

Let P and Q be the forces and P is greater than Q , acting at a point O inclined angle .Here P+Q = 18 ....(1) and the resultant  R =12 N.   Construct a parallelogram with sides with OA = Q, OB = P, with angle AOB as an obtuse angle and OA along the x axis.Draw the diagonal OC  to the parellelogram OACB  so that angle AOC =90 degree .The diagonal OC = resultant R which is perpendicular to the s maller force Q .  consider the triangle AOC , OA = P, AC = OB  = P, and angle AOC =90degree.That is triangle AOC is a right angled triangle with AC = P as hypotenuse. Then   P^ 2  = Q^2 +R^2  or P^2 - Q^2 = R^2  or (P+Q) (P - Q)  = R^2   or P -Q  = R^2/P+Q that is   P- Q  = 144/18   = 8  ...(2)    Adding (1) and (2)   2P = 26 or P = 13 N . Then Q = 5 N.

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