The main objective of this question is to resolve the given vector into its component and determine its magnitude.
This question uses the concept of Vector resolution. A vector resolution is the breaking of such a single vector into several vectors in various directions that collectively generate the same effect as a single vector. Component vectors are the vectors created following splitting.
Expert Answer
We have to resolve the given vectors into its component.
By using the sine rule, we get:
\[ \space \frac{F_2}{sin \space 70} \space = \space \frac{(F_2)_u}{sin \space 45} \space = \frac{(F_2)_v}{sin 65 } \]
Now calculating $ F_2 $ in the direction of $ u $.
So:
\[ \space \frac{F_2}{sin \space 70 } \space = \space \frac{(F_2)_u}{sin \space 45} \]
\[ \space (F_2)_u \space = \space \frac{F_2 \space \times \space sin \space 45 } {sin \space 70} \]
By putting the value of $F_2$, we get:
\[ \space (F_2)_u \space = \space \frac{500 \space \times \space sin \space 45 } {sin \space 70} \]
By simplifying, we get:
\[ \space (F_2)_u \space = \space 376.24 \]
Now resolving in the $ v $ direction.
\[ \space \frac{F_2}{sin \space 70 } \space = \space \frac{(F_2)_v}{sin \space 65} \]
\[ \space (F_2)_v \space = \space \frac{F_2 \space \times \space sin \space 65 } {sin \space 70} \]
By putting the value of $F_2$, we get:
\[ \space (F_2)_v \space = \space \frac{500 \space \times \space sin \space 65 } {sin \space 70} \]
By simplifying, we get:
\[ \space (F_2)_u \space = \space 482.24 \space N \]
Now magnitude is calculated as:
\[ \space F_2 \space = \space \sqrt{(F_2)^2_u \space + \space (F_2)^2_v} \]
By putting values, we get:
\[ \space = \space \sqrt {(376.24)^2 \space + \space (482.24)^2 } \]
\[ \space F_2 \space = \space 611.65 \space N \]
Numerical Answer
The magnitude of $ F_2 $ resolving into components is:
\[ \space F_2 \space = \space 611.65 \space N \]
Example
In the above question, if the magnitude of $ F_2 $ is $ 1000 \space N $, find the magnitude of $F_2$ after resolving into its components $u$ and $v$.
By using the sine rule, we get:
\[ \space \frac{F_2}{sin \space 70} \space = \space \frac{(F_2)_u}{sin \space 45} \space = \frac{(F_2)_v}{sin 65 } \]
Now calculating $ F_2 $ in the direction of $ u $.
So:
\[ \space \frac{F_2}{sin \space 70 } \space = \space \frac{(F_2)_u}{sin \space 45} \]
\[ \space (F_2)_u \space = \space \frac{F_2 \space \times \space sin \space 45 } {sin \space 70} \]
By putting the value of $F_2$, we get:
\[ \space (F_2)_u \space = \space \frac{1000 \space \times \space sin \space 45 } {sin \space 70} \]
By simplifying, we get:
\[ \space (F_2)_u \space = \space 752.48 \]
Now resolving in the $ v $ direction.
\[ \space \frac{F_2}{sin \space 70 } \space = \space \frac{(F_2)_v}{sin \space 65} \]
\[ \space (F_2)_v \space = \space \frac{F_2 \space \times \space sin \space 65 } {sin \space 70} \]
By putting the value of $F_2$, we get:
\[ \space (F_2)_v \space = \space \frac{1000 \space \times \space sin \space 65 } {sin \space 70} \]
By simplifying, we get:
\[ \space (F_2)_u \space = \space 964.47 \space N \]
Now magnitude is calculated as:
\[ \space F_2 \space = \space \sqrt{(F_2)^2_u \space + \space (F_2)^2_v} \]
By putting values, we get:
\[ \space = \space \sqrt {(752.48)^2 \space + \space (964.47)^2 } \]
\[ \space F_2 \space = \space 1223.28 \space N \]