Question

Two forces, each of magnitude $F$ have a resultant of magnitude $F$ . The angle between the two forces is
A. $45^\circ$
B. $120^\circ$
C. $150^\circ$
D. $60^\circ$

Answer 1

force is the push or pull applied on an object that causes the acceleration of the body. As given in the question, force is the vector. Therefore, we will use the method of resultant of vector quantities to find the angle between the two forces. Here, the resultant is also of the same magnitude as that of the forces.

Formula used:
The formula of the resultant vectors of two vector quantities is given by
$R = \sqrt {{A^2} + {B^2} + 2AB\cos \theta }$
Here, $R$ is the resultant, $A$ and $B$ are the vector quantities, and $\theta$ is the angle between these vector quantities.

Complete step by step answer:
Force is the push or pull applied on an object that causes the acceleration of the body. As we know that the force is the vector. Therefore, we will use the method of resultant vector quantities. Therefore, the resultant of the two vectors $A$ and $B$ is given by
$R = \sqrt {{A^2} + {B^2} + 2AB\cos \theta }$
Here, $\theta$ is the angle between the vectors $A$ and $B$ .
Now, it is given in the question that, there are two forces each of magnitude $F$ and their resultant is also of magnitude $F$ . Therefore, we will put $F$ in place of $A$ , $B$ and $R$ .
Therefore, the resultant $F$ of the two vectors forces each of magnitude $F$ is given by
$F = \sqrt {{F^2} + {F^2} + 2F.F\cos \theta }$
$\Rightarrow \,F = \sqrt {2{F^2} + 2{F^2}\cos \theta }$
$\Rightarrow \,F = \sqrt {2{F^2}\left( {1 + \cos \theta } \right)}$
Now, squaring both the sides, we get
${F^2} = 2{F^2}\left( {1 + \cos \theta } \right)$
$\Rightarrow \,\dfrac{{{F^2}}}{{2{F^2}}} = 1 + \cos \theta$
$\Rightarrow \,\dfrac{1}{2} = 1 + \cos \theta$
$\Rightarrow \,\dfrac{{ - 1}}{2} = \cos \theta$
$\therefore \,\theta = 120^\circ$
Therefore, the angle between the two forces is $120^\circ$ .

Hence, option B is the correct option.

Note: From the above result, we can say that if the forces have the same magnitude and same resultant, then the angle between the two forces will be $120^\circ$ . We can also say that the resultant is directed exactly halfway between the original two vectors.Therefore, the angle between the original two vectors and the resultant is half the angle between the original two vectors.