The sum of the magnitudes of two forces acting at a point . . .

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

Doubt by Suhani

Solution :

R=12 N

A+B=18 (Given) B=18-A — (1)

tanβ=Bsinθ/(A+Bcosθ) tan90°=Bsinθ/(A+Bcosθ) 1/0=Bsinθ/(A+Bcosθ) A+Bcoθ = 0 × Bsinθ

A+Bcosθ=0 Bcosθ=-A — (2)

R²=A²+B²+2ABcosθ R²=A²+(18-A)²+2A(-A) [Using Equation (1) and (2)] R²=A²+(18)²+A²-2(18)(A)-2A² (12)²=2A²+324-36A-2A² 144-324=-36A -180=-36A 180/36 = A 5=A A=5 N

Putting in equation (1)

B=18-A B=18-5 B=13 N

Hence, the magnitude of two forces are 5 N and 13 N.

Similar Question : The simple sum of two co-initial vectors is 16 units. Their vector sum is 8 units. The resultant of the vectors is perpendicular to the smaller vector. The magnitudes of the two vectors are a) 2 units and 14 units b) 4 units and 12 units c) 6 units and 10 units d) 8 units and 8 units.