Question

The sum of the magnitudes of two forces acting at a point is $18N$ and the magnitude of their resultant is $12N$. If the resultant makes an angle of $90^\circ $ with the force of smaller magnitude, what are the magnitude of two forces?

Answer 1

Let the two individual forces acting at a point is given as $\overrightarrow A $and $\overrightarrow B $and $\theta $ be the angle between the two forces $\overrightarrow A $and $\overrightarrow B $let$A < B$. If the resultant $\overrightarrow R $ makes an angle $\beta $ with the force $\overrightarrow A $ then
$\tan \beta = \dfrac{{B\sin \theta }}{{A + B\cos \theta }}$

As we taken the resultant angle $\beta $ that is $90^\circ $

$\tan 90 = \dfrac{{B\sin \theta }}{{A + B\cos \theta }}$ or $A + B\cos \theta = 0$

Complete step by step solution:
As we taken in the hint two individual forces acting at a point is given as $\overrightarrow A $and $\overrightarrow B $so]
$A + B = 18N$

Therefore we know the formulae for the resultant $R$ hence the equation will be

$R = \sqrt {{A^2} + {B^2} + 2AB\cos \theta } = 12$

Now we have to shift the root on the R.H.S so we get

${A^2} + {B^2} + 2AB\cos \theta = 144.............(1)$

Now we have to take the value of $B$ and $B\cos \theta $

So,$A + B = 18$

Now shift the $A$ on the L.H.S so we get

$B = 18 - A$

Now we have the equation $A + B\cos \theta = 0$

Now we want value of $B\cos \theta $ hence we get

$B\cos \theta = - A$

Now substitute the value of $B$ and $B\cos \theta $ in the equation (1)

${A^2} + {\left( {18 - A} \right)^2} + 2A\left( { - A} \right) = 144$

Here ${\left( {18 - A} \right)^2}$ is in the form of ${\left( {a - b} \right)^2}$ so we have to
apply that formulae

${A^2} + \left( {{{18}^2} + {A^2} - 2\left( {18} \right)\left( A \right)} \right) + 2A\left( { - A} \right) = 144$

After simplifying the above equation we get

${A^2} + 324 + {A^2} - 36A - 2{A^2} = 144$

Now we have to do further calculation

$2{A^2} - 36A - 2{A^2} = 144 - 324$

Now $2{A^2}$ and $ - 2{A^2}$ get cancelled so we get
$ - 36A = - 180$

Now we want the value of $A$ so

$A = \dfrac{{ - 180}}{{ - 36}}$

Hence $A = 5N$

And we have the equation,$A + B = 18$

We got the value of $A$ so substitute the value of $A$ in the above equation

$5 + B = 18$

Now we want the value of $B$ so we get

$B = 18 - 5$

Hence the value of $B = 13N$

Hence the values are $A = 15N$ and $B = 13N$

Note: Magnitude generally refers to the quantity or a distance. In relation to the movement, we correlate the magnitude with the size and the speed of the object while moving.