Question

The resultant of two forces $3P$ and $2P$ is $R$ if the first force is doubled then the resultant is also doubled. The angle between the two forces is
(A) ${60^ \circ }$
(B) ${120^ \circ }$
(C) ${70^ \circ }$
(D) ${180^ \circ }$

Answer 1

Use the given two forces and form the two equations with the resultant forces. Arrange the two equations in such a way that they can be solved by each other. By solving the equations, get the value of the cosine angle between the forces and from that take inverse to find the answer.

Complete step by step answer:
Given: The first force is $3P$
The second force is $2P$
The resultant force is $R$
If the first force doubles then the resultant force also doubles.
The first equation is ${R^2} = 9{P^2} + 4{P^2} + 12{P^2}\cos \theta$
By simplifying the above equation , we get
${R^2} = 13{P^2} + 12{P^2}\cos \theta$ ------------(1)
By getting the second equation of the resultant forces,
${\left( {2R} \right)^2} = {\left( {2 \times 3P} \right)^2} + 4{P^2} + 24{P^2}\cos \theta$
By simplifying the above step, we get
${\left( {2R} \right)^2} = {\left( {6P} \right)^2} + 4{P^2} + 24{P^2}\cos \theta$
By further simplification,
$4{R^2} = {\left( {6P} \right)^2} + 4{P^2} + 24{P^2}\cos \theta$ -------(2)
In order to equate the equation (1) and (2), multiply the equation (1) into $4$
$4{R^2} = 52{P^2} + 48{P^2}\cos \theta$ --------(3)
Solving the equation (2) and (3), we get
${\left( {6P} \right)^2} + 4{P^2} + 24{P^2}\cos \theta = 52{P^2} + 48{P^2}\cos \theta$
By doing the addition and the subtraction at a necessary point to do the simplification,
$12{P^2} = - 24{P^2}\cos \,\theta$
$\cos \theta = \dfrac{{12{P^2}}}{{24{P^2}}}$
By cancelling the similar terms in the right hand side of the equation,
$\cos \theta = - \dfrac{1}{2}$
$\Rightarrow \theta = {\cos ^{ - 1}}\left( {\dfrac{1}{2}} \right)$.

Hence, the angle between the two forces is ${120^ \circ }$.

Note: The force is said to be vector, if it possesses both magnitude and direction. The angle between the forces matters much, this is because the two forces can act in two different directions, the resultant must be in the direction of the average of the two directions.