Question

The sum of magnitudes of two forces acting at a point is 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

Here, we should use the triangular law of vector addition. Since the angle between the resultant and smaller force is $90^\circ$, we understand that the three forces will constitute a right angled triangle and the Pythagoras theorem can be applied to find the force magnitudes.
Formulae used: For right-angled triangle, $hypotenus{e^2} = bas{e^2} + altitud{e^2}$

Complete step-by-step answer:
Let us assume that the magnitude of the force with a smaller magnitude is ${F_1}$, the magnitude of the second force is ${F_2}$ and the magnitude of the resultant force is ${F_R}$.
According to the question,
${F_1} + {F_2} = 18$
and
${F_R} = 12$
Since the magnitudes of forces ${F_1}$, ${F_2}$ and the resultant ${F_R}$ form sides of a right-angled triangle, from the Pythagoras theorem, we can write as:
$F_1^2 + F_R^2 = F_2^2$
Rearranging the above equation we get,
$F_R^2 = F_2^2 - F_1^2 = {12^2} = 144$ ......equation(1)
Rearranging the equation ${F_1} + {F_2} = 18$ we get,
${F_1} = 18 - {F_2}$ ......equation(2)
Now, substituting (2) in (1), we get
$F_2^2 - \left[ {{{\left( {18 - {F_2}} \right)}^2}} \right] = 144$
On simplifying further we get,
$F_2^2 - {18^2} + 36{F_2} - F_2^2 = 144$
Now, solving the equation to get the value of ${F_2}$ as:
$36{F_2} = 144 + {18^2} = 144 + 324 = 468$
Dividing both sides by 36 we get,
$\therefore {F_2} = \dfrac{{468}}{{36}} = 13$
Hence, the magnitude of the stronger force is ${F_2} = 13$.
This value can be substituted in equation(2) to get the value of the weak force (force with smaller magnitude), that is:
${F_1} = 18 - {F_2} = 18 - 13 = 5$

Therefore, we obtained the magnitudes of the two forces as: 5 and 13 and the correct answer is option C.

Note: Since force is a vector quantity and also because force vectors can be arranged as the two sides of a triangle in sequence, the third side in the opposite sequence represents the resultant of the forces. This is the law of triangles for the vector addition. While performing any problem using the triangle law of vector addition, the direction of the vectors in sequence must be taken care.