Question

The sum of the magnitudes of two forces acting at a point is 18N and the magnitude of their resultant is 12N.If the resultant makes an angle of 90° with the force of the smaller magnitude,what are the magnitude of two forces?

Answer 1

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.