Question

The sum of the magnitudes of two forces acting at a point is $18$ and the magnitude of their resultant is $12$. If the resultant is at $90°$ with a force of smaller magnitude, what are the magnitudes of forces?

• A. $2,5$
• B. $14,4$
• C. $5,13$
• D. $10,8$

Answer 1

The correct option is C

$5,13$

Step 1. Given Data,

The sum of magnitudes of two forces is given as,

$\left|P\right|+\left|Q\right|=18$

The magnitude of their resultant is 12,

$Q^2-P^2={12}^2$

Step 2. Formula Used,

The resultant force of $\overrightarrow{P}$ and $\overrightarrow{Q}$ is $\overrightarrow{R}=\overrightarrow{P}+\overrightarrow{Q}$

$R=P²+Q²+2PQCosФ$

$\overrightarrow{P}$and $\overrightarrow{Q}$ be the two vectors.

Let the angle between the two vectors be $Ф$

Let $P$ be the magnitude of the smaller force, $Q$ be the magnitude of the larger force and the resultant force be $R$.

Thus, it can be written as,

$Q^2-P^2=R^2$

Step 3. Calculate the magnitude of Forces,

Let the magnitude of the smaller force is $P$, the magnitude of the larger force is $Q$ and the resultant force be $R$.

As the resultant force makes ${90}^o$ with the smaller force $P$ then $Q$ forms the hypotenuse of the triangle.

Thus, it can be written as,

$Q^2-P^2=R^2$

$Q^2-P^2={12}^2$

$Q^2-P^2=144......\left(1\right)$

The sum of magnitudes of two forces is given as,

$P+Q=18$

$Q=18-P......\left(2\right)$

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 are $5N$ and $13N$.

Hence option C is the correct answer.