Question

Two forces with equal magnitudes F act on a body and the magnitude of the resultant force is $\dfrac{F}{3}$ . The angle between the two forces is:
(A) ${\cos ^{ - 1}}\left( {\dfrac{1}{{2\sqrt 3 }} - 1} \right)$
(B) ${\cos ^{ - 1}}\left( { - \dfrac{1}{3}} \right)$
(C) ${\cos ^{ - 1}}\left( {\dfrac{1}{{2\sqrt 3 }} + 1} \right)$
(D) ${\cos ^1}\left( { - \dfrac{8}{9}} \right)$

Answer 1

Hint:First start with what is the resultant force on a body and how it is calculated. For this you need to know the relation between forces applied on a body and the resultant force of all those forces applied on the body. Use the relation and put all the given values from the question into that relation.

Formula used:
The expression of resultant force is,
${F_R} = \sqrt {{F_1}^2 + {F_2}^2 + 2{F_1}{F_2}\cos \theta } $
Here, $F_1,F_2$ are the forces and $\cos \theta$ is the angle between forces $F_1$ and $F_2$.

Complete step by step solution:
We know that the resultant force on a body is equal to the total amount of force acting on the body along with the direction of the force applied on the body. Here in the question two forces applied on a body which is of equal magnitude F. Let the two forces be ${F_1}\,and\,{F_2}$. Let the resultant force be ${F_R} = \dfrac{F}{3}$ (as given in the question).

Now we know the formula of resultant force on a body is as follows-
${F_R} = \sqrt {{F_1}^2 + {F_2}^2 + 2{F_1}{F_2}\cos \theta } \\$
Substituting the value of resultant force and the two forces applied on the body in the above equation we get-
$\dfrac{F}{3} = \sqrt {{F^2} + {F^2} + 2FF\cos \theta } \\$
After further solving we get,
$\dfrac{{{F^2}}}{{\sqrt 3 }} = 2{F^2}\left( {1 + \cos \theta } \right) \\ $
Now try to find out the value of $\cos \theta $ from the above equation.
$\cos \theta = \left( {\dfrac{1}{{2\sqrt 3 }} - 1} \right)$
Hence, $\theta = {\cos ^{ - 1}}\left( {\dfrac{1}{{2\sqrt 3 }} - 1} \right)$
Where, $\theta $ is the angle between the two forces of same magnitude applied on the body.

Hence, the correct answer is Option A.

Note: Here the two forces applied were of same magnitude but it is not the case all the time. So read the given information carefully before starting putting the values in the formula. Also when the force applied is parallel or perpendicular then this formula will not be used. As in that case the value of angle will be zero.