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Two forces have magnitude in the ratio 5:12. If the angle between them is 90°, they give the resultant of magnitude 26N. Find the magnitude of resultant force if the angle between them is changed to 120°.

Let magnitude of two forces be F1and F2.

Here,      straight F subscript 1 over straight F subscript 2 equals 5 over 12 

rightwards double arrow      space space space straight F subscript 2 equals 12 over 5 straight F subscript 1 equals 2.4 straight F subscript 1     ...(1)

The resultant of stack straight F subscript 1 with rightwards arrow on top space and space stack straight F subscript 2 with rightwards arrow on top is 26N when angle between them is .

Therefore,

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                      equals straight F subscript 1 squared plus straight F subscript 2 squared              ...(2)

Substituting, straight F subscript 2 equals 2.4 space straight F subscript 1 in equation (2),

      left parenthesis 26 right parenthesis squared equals straight F subscript 1 squared plus straight F subscript 2 squared equals straight F subscript 1 squared plus left parenthesis 2.4 right parenthesis squared straight F subscript 1 squared

space space space space space space space space space equals 6.76 straight F subscript 1 squared
 
     left parenthesis 26 right parenthesis squared equals left parenthesis 2.6 right parenthesis squared straight F subscript 1 squared

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∴  straight F subscript 2 equals 2.4 space space space space space straight F subscript 1 equals 24 straight N

Let R be the magnitude of the resultant when the angle between the two forces is 120 degree comma

∴                    straight R squared equals straight F subscript 1 squared plus straight F subscript 2 squared plus 2 straight F subscript 1 straight F subscript 2 space cos space 120 degree

                          equals straight F subscript 1 squared plus straight F subscript 2 squared minus straight F subscript 1 straight F subscript 2

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i.e.,                   straight R equals 20.88, is the magnitude of the resultant between the vectors. 
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