Question

The sum of the magnitudes of two forces acting at point us 18 and the magnitude of their resultant is 12. If the resultant is at $90{}^\circ$ with the force of smaller magnitude, what are the, magnitudes of forces
(A) 12.5
(B) 14.4
(C) 5.13
(D) 10.8

Answer 1

Hint: We know that a force is a push or pull upon an object resulting from the object’s interaction with another object. Whenever there is an interaction between two objects, there is a force upon each of the objects. When the interaction ceases, the two objects no longer experience the force. Based on the concept we have to answer this question.

Complete step by step answer:
Let us consider that the magnitude of smaller force is P, the magnitude of larger force Q and the resultant force is R.
As the resultant force makes $90{}^\circ$with the smaller force P then Q forms the hypotenuse of the triangle.
Thus, it can be written as,
${{Q}^{2}}-{{P}^{2}}={{R}^{2}}$
$\ {{Q}^{2}}-{{P}^{2}}={{(12)}^{2}}$
$\ {{Q}^{2}}-{{P}^{2}}=144$ ……………………….. equation (1)
The sum of magnitudes of two forces is given as,
$P+Q=18$
$Q=18-P$ ……………………….. equation (2)
Substituting the value from equation (2) in equation (1), we get
${{\left( 18-P \right)}^{2}}-{{P}^{2}}=144$
$P=5N$
Substituting the value of P in equation (1), we get
$Q=18-5\ $
$=13N$
Thus, the magnitude of the forces is 5 N and 13 N.

Hence, the correct answer is Option C.

Note: A resultant force, as known by us is defined as the single force and associated torque obtained by combining a system of forces and torques acting on a rigid body. The defining feature of a resultant force, or resultant force or torque, is that it has the same effect on the rigid body as the original system of forces.
To calculate the resultant force, we can use that the parallelogram of the force’s method is one of the graphical methods developed to find the resultant of a coplanar force system. Two or more concurrent forces can be replaced by a single resultant force that is statically equivalent to these forces.