Question

Two forces, with equal magnitude F, act on a body and the magnitude of the resultant force is $\frac { F } { 3 }$ . The angle between the two forces is

• A. $\cos ^ { - 1 } \left( - \frac { 17 } { 18 } \right)$
• B. $\cos ^ { - 1 } \left( - \frac { 1 } { 3 } \right)$
• C. $\cos ^ { - 1 } \left( \frac { 2 } { 3 } \right)$
• D. $\cos ^ { - 1 } \left( \frac { 8 } { 9 } \right)$

Answer 1

Answer: (A)

$\cos ^ { - 1 } \left( - \frac { 17 } { 18 } \right)$

Answer 2

Resultant of two vectors A and B, which are working at an angle $\theta$ , can be given by

$R = \sqrt { A ^ { 2 } + B ^ { 2 } + 2 A B \cos \theta }$

[As $A = B = F$ and $R = \frac { F } { 3 }$ ]

$\left( \frac { F } { 3 } \right) ^ { 2 } = F ^ { 2 } + F ^ { 2 } + 2 F ^ { 2 } \cos \theta$ $\frac { F ^ { 2 } } { 9 } = 2 F ^ { 2 } + 2 F ^ { 2 } \cos \theta$ ⇒

$\frac { - 17 } { 9 } F ^ { 2 } = 2 F ^ { 2 } \cos \theta$ ⇒ $\cos \theta = \left( \frac { - 17 } { 18 } \right)$ or

$\theta = \cos ^ { - 1 } \left( \frac { - 17 } { 18 } \right)$