Question

Two forces, F₁ and F₂ are acting on a body. One force is double that of the other force and the resultant is equal to the greater force. Then the angle between the two forces is

• A. $A={\cos }^{-1}\left(\frac{1}{2}\right)$
• B. $A={\cos }^{-1}\left(\frac{-1}{2}\right)$
• C. $A={\cos }^{-1}\left(\frac{-1}{4}\right)$
• D. $A={\cos }^{-1}\left(\frac{1}{4}\right)$

Answer 1

The correct option is C

$A={\cos }^{-1}\left(\frac{-1}{4}\right)$

Step 1: Given;

$F_1=F$;

$F_2=2F$;

$F_R=2F$

Step 2: Formula used:

$F_R=\sqrt{F_1^2+F_2^2+2F_1{F}_2\cos A}$ ($F_R$ is the resultant force, $F_1,F_2$ are the two forces acting, $A$ is the angle between two forces)

Step 3: Find the angle,$A$ between the two forces;

$$\begin{gathered} 2F=\sqrt{F^2+4F^2+4F^2\cos A} \\ -F^2=4F^2\cos A \\ cosA=\frac{-1}{4} \\ A={\cos }^{-1}\left(\frac{-1}{4}\right) \end{gathered}$$

Hence the angle between two forces is, $A={\cos }^{-1}\left(\frac{-1}{4}\right)$.

Thus, option C is correct.