question_answer

The sum of the magnitudes of two forces acting at a point is 18 N and the magnitude of their resultant is 12N. If the resultant is at \[{{90}^{{}^\circ }}\]with the smaller force, the magnitude of the forces in N are

A)  6, 12                                     

B)  11, 7     

C)         5, 13                     

D)         14, 4

E)  10, 8   

Correct Answer: C

Solution :

Let smaller force be\[{{F}_{1}}\]. Resultant R of the forces is at\[90{}^\circ \]to AB, \[\therefore \,\,{{R}^{2}}\,+F_{1}^{2}\,=F_{2}^{2}\] in \[\Delta \,ABC\] or            \[{{(12)}^{2}}=F_{2}^{2}-F_{1}^{2}\]                       ...(i) or            \[144=({{F}_{2}}-{{F}_{1}})({{F}_{2}}-{{F}_{1}})\] but         \[{{F}_{1}}+{{F}_{2}}=18\,N\]                      (given)... (ii) \[\therefore \]  \[{{F}_{2}}-{{F}_{1}}=\frac{144}{18}=8\]       ??(iii) From Eqs. (ii) and (iii), \[{{F}_{1}}=5,\text{ }{{F}_{2}}=13\] Hence, forces are 5 N and 13 N.