# 10.3 Components  (Page 2/3)

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## Worked example

Determine the force needed to keep a 10 kg block from sliding down a frictionless slope. The slope makes an angle of 30 ${}^{\circ }$ with the horizontal.

1. The force that will keep the block from sliding is equal to the parallel component of the weight, but its direction is up the slope.

2. $\begin{array}{ccc}\hfill {F}_{g\parallel }& =& mgsin\theta \hfill \\ & =& \left(10\right)\left(9,8\right)\left(sin{30}^{\circ }\right)\hfill \\ & =& 49\mathrm{N}\hfill \end{array}$
3. The force is 49 N up the slope.

## Vector addition using components

Components can also be used to find the resultant of vectors. This technique can be applied to both graphical and algebraic methods of finding the resultant. The method is simple: make a rough sketch of the problem, find the horizontal and vertical components of each vector, find the sum of all horizontal components and the sum of all the vertical components and then use them to find the resultant.

Consider the two vectors, $\stackrel{\to }{A}$ and $\stackrel{\to }{B}$ , in [link] , together with their resultant, $\stackrel{\to }{R}$ .

Each vector in [link] can be broken down into one component in the $x$ -direction (horizontal) and one in the $y$ -direction (vertical). These components are two vectors which when added give you the original vector as the resultant. This is shown in [link] where we can see that:

$\begin{array}{ccc}\hfill \stackrel{\to }{A}& =& {\stackrel{\to }{A}}_{x}+{\stackrel{\to }{A}}_{y}\hfill \\ \hfill \stackrel{\to }{B}& =& {\stackrel{\to }{B}}_{x}+{\stackrel{\to }{B}}_{y}\hfill \\ \hfill \stackrel{\to }{R}& =& {\stackrel{\to }{R}}_{x}+{\stackrel{\to }{R}}_{y}\hfill \end{array}$
$\begin{array}{ccc}\hfill \mathrm{But},\phantom{\rule{1.em}{0ex}}{\stackrel{\to }{\mathrm{R}}}_{\mathrm{x}}& =& {\stackrel{\to }{A}}_{x}+{\stackrel{\to }{B}}_{x}\hfill \\ \hfill \mathrm{and}\phantom{\rule{1.em}{0ex}}{\stackrel{\to }{\mathrm{R}}}_{\mathrm{y}}& =& {\stackrel{\to }{A}}_{y}+{\stackrel{\to }{B}}_{y}\hfill \end{array}$

In summary, addition of the $x$ components of the two original vectors gives the $x$ component of the resultant. The same applies to the $y$ components. So if we just added all the components together we would get the same answer! This is another importantproperty of vectors.

If in [link] , $\stackrel{\to }{A}=5,385m·{s}^{-1}$ at an angle of 21.8 ${}^{\circ }$ to the horizontal and $\stackrel{\to }{B}=5m·{s}^{-1}$ at an angle of 53,13 ${}^{\circ }$ to the horizontal, find $\stackrel{\to }{R}$ .

1. The first thing we must realise is that the order that we add the vectors does not matter. Therefore, we can work through the vectors to be added in any order.

2. We find the components of $\stackrel{\to }{A}$ by using known trigonometric ratios. First we find the magnitude of the vertical component, ${A}_{y}$ :

$\begin{array}{ccc}\hfill sin\theta & =& \frac{{A}_{y}}{A}\hfill \\ \hfill sin21,{8}^{\circ }& =& \frac{{A}_{y}}{5,385}\hfill \\ \hfill {A}_{y}& =& \left(5,385\right)\left(sin21,{8}^{\circ }\right)\hfill \\ & =& 2m·{s}^{-1}\hfill \end{array}$

Secondly we find the magnitude of the horizontal component, ${A}_{x}$ :

$\begin{array}{ccc}\hfill cos\theta & =& \frac{{A}_{x}}{A}\hfill \\ \hfill cos21.{8}^{\circ }& =& \frac{{A}_{x}}{5,385}\hfill \\ \hfill {A}_{x}& =& \left(5,385\right)\left(cos21,{8}^{\circ }\right)\hfill \\ & =& 5m·{s}^{-1}\hfill \end{array}$

The components give the sides of the right angle triangle, for which the original vector, $\stackrel{\to }{A}$ , is the hypotenuse.

3. We find the components of $\stackrel{\to }{B}$ by using known trigonometric ratios. First we find the magnitude of the vertical component, ${B}_{y}$ :

$\begin{array}{ccc}\hfill sin\theta & =& \frac{{B}_{y}}{B}\hfill \\ \hfill sin53,{13}^{\circ }& =& \frac{{B}_{y}}{5}\hfill \\ \hfill {B}_{y}& =& \left(5\right)\left(sin53,{13}^{\circ }\right)\hfill \\ & =& 4m·{s}^{-1}\hfill \end{array}$

Secondly we find the magnitude of the horizontal component, ${B}_{x}$ :

$\begin{array}{ccc}\hfill cos\theta & =& \frac{{B}_{x}}{B}\hfill \\ \hfill cos21,{8}^{\circ }& =& \frac{{B}_{x}}{5,385}\hfill \\ \hfill {B}_{x}& =& \left(5,385\right)\left(cos53,{13}^{\circ }\right)\hfill \\ & =& 5m·{s}^{-1}\hfill \end{array}$

4. Now we have all the components. If we add all the horizontal components then we will have the $x$ -component of the resultant vector, ${\stackrel{\to }{R}}_{x}$ . Similarly, we add all the vertical components then we will have the $y$ -component of the resultant vector, ${\stackrel{\to }{R}}_{y}$ .

$\begin{array}{ccc}\hfill {R}_{x}& =& {A}_{x}+{B}_{x}\hfill \\ & =& 5m·{s}^{-1}+3m·{s}^{-1}\hfill \\ & =& 8m·{s}^{-1}\hfill \end{array}$

Therefore, ${\stackrel{\to }{R}}_{x}$ is 8 m to the right.

$\begin{array}{ccc}\hfill {R}_{y}& =& {A}_{y}+{B}_{y}\hfill \\ & =& 2m·{s}^{-1}+4m·{s}^{-1}\hfill \\ & =& 6m·{s}^{-1}\hfill \end{array}$

Therefore, ${\stackrel{\to }{R}}_{y}$ is 6 m up.

5. Now that we have the components of the resultant, we can use the Theorem of Pythagoras to determine the magnitude of the resultant, $R$ .

$\begin{array}{ccc}\hfill {R}^{2}& =& {\left({R}_{x}\right)}^{2}+{\left({R}_{y}\right)}^{2}\hfill \\ \hfill {R}^{2}& =& {\left(6\right)}^{2}+{\left(8\right)}^{2}\hfill \\ \hfill {R}^{2}& =& 100\hfill \\ \hfill \therefore R& =& 10m·{s}^{-1}\hfill \end{array}$

The magnitude of the resultant, $R$ is 10 m. So all we have to do is calculate its direction. We can specify the direction as the angle the vectors makes with a known direction. To do this you only need to visualise the vector as starting at the origin of a coordinate system. We have drawn this explicitly below and the angle we will calculate is labeled $\alpha$ .

Using our known trigonometric ratios we can calculate the value of $\alpha$ ;

$\begin{array}{ccc}\hfill tan\alpha & =& \frac{6m·{s}^{-1}}{8m·{s}^{-1}}\hfill \\ \hfill \alpha & =& {tan}^{-1}\frac{6m·{s}^{-1}}{8m·{s}^{-1}}\hfill \\ \hfill \alpha & =& 36,{8}^{\circ }.\hfill \end{array}$
6. $\stackrel{\to }{R}$ is 10 m at an angle of $36,{8}^{\circ }$ to the positive $x$ -axis.

#### Questions & Answers

How do hydrogen and chlorine atoms bond covalently in a molecule of hydrogen chloride?
Tiny Reply
what is the parallel circuit
Philani Reply
how many moles of H2O can be formed if 12,5 moi CH4 reacts with sufficient NH3 and O2
Nomcebo Reply
what atoms form covalent bond
hehe Reply
The combination of two non metals
Ndumiso
combined two non-metals 😊
Trudy
what is a mole
Thandeka Reply
a mole is a measure of large quantities of small entities such as atoms, molecules etc
DINEO
what does STP stand for?
Deeco
STP stands for Standard Temperature and Pressure.
Dolly
A car drives straight off the edge of a cliff that is 54m high. The police at the scene of the accident observed that the point of the impact is 130m from the base of the clif. Calculate the initial velocity of the car when it went ovet the clif.
hanyani Reply
wat happens to current if resistors are in parallel connection
Mosima Reply
more current flows from the source that would flow for any of them individually,so the total total resistance is lower
Theoo
In the parallel circuit the current is divided among the resistors
Ndumiso
what are the isotopes
Tlotlisang Reply
are different types of the same elemant but with different mass or atomic no.
Ayanda
i sotopes are different type of element with same atomic number but different mass number
Tareskay
yes i forgot some details😁
Ayanda
next quetion plz
Tareskay
why don't we insert the negative sign for 5 × 10 - 9 when substituting
Mpho Reply
Why does an enclosed gas exert pressure on the walls of a container
Palesa Reply
State the gay lussacs law
Anna Reply
what is the coefficient of Na in order to balanced the equation?_Na + MgCl2=2NaCl+Mg?
Arcel Reply
2
Okuhle
2
Tlotlisang
2
Praise
2
Anna
2
Anelehr
2
Liyema
the combining power of an element, especially as measured by the number of hydrogen atoms it can displace or combine with.
Dealon Reply
Thanks
REJOYCE
download periodic table from play Store...it will explain everything to u
Dealon
what are orbitals
Sphe Reply
examples of atoms whose Valence energy levels are not full and more to bond and become more stable
Sphe
What is a valency
REJOYCE
What a lone pairs...
Ndumiso
Valency is the number of electrons than an atom must gain, lose or share to achieve noble gas configuration.
Mcebisi
what is vacuum
Njabulo Reply
Vacuum, space in which there is no matter or in which the pressure is so low that any particles in the space do not affect any processes being carried on there. It is a condition well below normal atmospheric pressure and is measured in units of pressure (the pascal).
Khayalethu

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Source:  OpenStax, Siyavula textbooks: grade 11 physical science. OpenStax CNX. Jul 29, 2011 Download for free at http://cnx.org/content/col11241/1.2
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