Question

velocity? 6. A force 7 of magnitude 12 N has non-rectangular components and the sum of the magnitudes of P and 0 is 18 N. The direction of Ő is at right angles to ĒFind the magnitude of 0. then takes an elevator, 12 m down​

Answer 1

Follow the given picture

Answer 2

The magnitude of O is 5 N.

A force is said to be a vector quantity which means that is involves both magnitude as well as direction.

Since, force is a vector is can be resolved into various other components. These components which when summed up would give the magnitude of  resultant force or the total force.

Usually force or any vector may be decomposed into 2 components namely the sine and the cosine components that are at right angles to one another and these components are the ones that are termed as rectangular components.

However, it is given that the force vector has a magnitude of 12 N which consists of non rectangular components which means that the components are not perpendicular to each other and are rather inclined to form angles other than 90°.

The two components that the force vector is split into are $\vec P$ and $\vec O$vectors that are inclined at some angle and do not form right angles with each other.

This can be seen from the diagram that shows the force vector and its components split.

(Diagram is attached at the end)

The same diagram can be resolved into a triangle where the vector $\vec P$   would be the resultant of vectors $\vec F$ and $\vec O$.

(Diagram is attached at the end)

The vector can be shifted to form this right angled triangle because it is given that the direction of $\vec O$ is placed in a manner that it is perpendicular to $\vec F$.

This triangle is formed as it the sum of magnitudes of vectors  $\vec F$ and $\vec O$ will be equivalent to the resultant vector $\vec P$.

Thus we have,

$F^{2} +O^{2} =P^{2}$

By rearranging the terms we have:

$\implies F^{2} =P^{2}-O^{2}$

By the properties used in algebra the above equation can be written as:

$\implies F^{2} =(P-O)(P+O)$   [∵ $a^{2}- b^{2} =(a+b)(a-b)$]

It is given that the magnitude of $\vec F$ is 12 N and the sum of the magnitudes of $\vec P$ and $\vec O$ is 18 N.

Hence by substituting these values we get:

$\implies 12^{2} =18\times (P+Q)$

Hence, the sum of their magnitudes will be:

$\implies (P+O)=\frac{144}{18}$

$(P+O)=8\; N ------(1)$

We already know that,

$(P-O)=18------(2)$

So by solving both equations (subtracting (2) from (1)) we have:

$\implies -2O=-10$

$O=5\;N$

Thus, this is the magnitude of vector $\vec O$.

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