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Resolve the force F2 into components acting along the u and v axes and determine the magnitudes of the components.

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:

\[ \space = \space \sqrt {(752.48)^2 \space + \space (964.47)^2 } \]

\[ \space F_2 \space = \space 1223.28 \space N \]

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