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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 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. F g = m g sin θ = ( 10 ) ( 9 , 8 ) ( sin 30 ) = 49 N
  3. The force is 49 N up the slope.

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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, A and B , in [link] , together with their resultant, R .

An example of two vectors being added to give a resultant

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:

A = A x + A y B = B x + B y R = R x + R y
But , R x = A x + B x and R y = A y + B y

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.

Adding vectors using components.

If in [link] , A = 5 , 385 m · s - 1 at an angle of 21.8 to the horizontal and B = 5 m · s - 1 at an angle of 53,13 to the horizontal, find 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 A by using known trigonometric ratios. First we find the magnitude of the vertical component, A y :

    sin θ = A y A sin 21 , 8 = A y 5 , 385 A y = ( 5 , 385 ) ( sin 21 , 8 ) = 2 m · s - 1

    Secondly we find the magnitude of the horizontal component, A x :

    cos θ = A x A cos 21 . 8 = A x 5 , 385 A x = ( 5 , 385 ) ( cos 21 , 8 ) = 5 m · s - 1

    The components give the sides of the right angle triangle, for which the original vector, A , is the hypotenuse.

  3. We find the components of B by using known trigonometric ratios. First we find the magnitude of the vertical component, B y :

    sin θ = B y B sin 53 , 13 = B y 5 B y = ( 5 ) ( sin 53 , 13 ) = 4 m · s - 1

    Secondly we find the magnitude of the horizontal component, B x :

    cos θ = B x B cos 21 , 8 = B x 5 , 385 B x = ( 5 , 385 ) ( cos 53 , 13 ) = 5 m · s - 1

  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, R x . Similarly, we add all the vertical components then we will have the y -component of the resultant vector, R y .

    R x = A x + B x = 5 m · s - 1 + 3 m · s - 1 = 8 m · s - 1

    Therefore, R x is 8 m to the right.

    R y = A y + B y = 2 m · s - 1 + 4 m · s - 1 = 6 m · s - 1

    Therefore, 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 .

    R 2 = ( R x ) 2 + ( R y ) 2 R 2 = ( 6 ) 2 + ( 8 ) 2 R 2 = 100 R = 10 m · s - 1

    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 α .

    Using our known trigonometric ratios we can calculate the value of α ;

    tan α = 6 m · s - 1 8 m · s - 1 α = tan - 1 6 m · s - 1 8 m · s - 1 α = 36 , 8 .
  6. R is 10 m at an angle of 36 , 8 to the positive x -axis.

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Questions & Answers

how to calculate emf when you are given two circuits with different external resistance, current but no internal resistance?
Andiswa Reply
you have to find the current of whole both circuits nd add em...
Amanzialwandle
calculate the distance between two charge +4nc and -3nc if electrostatic is 0.005N
Malaza Reply
use the coulombs law equation where there is F u substite by 0.005 then u cross multiple
Neldo
what is atomic
motheo Reply
what is charge
motheo
how to calculate the distance between Q1 and point P if you are given distance between point P and Q2
Lwando Reply
how do we calculate the magnitude of the force between two objects
Ntombi Reply
we use Newtons law ...fnet=ma
Fikile
how do you calculate the enery change in enthalpy reaction, can someone explain it to me guys
Buyie Reply
∆H(enthalpy change)= Hproducts- Hreactants.... use this formula, H stands for enthalpy which is regarded as energy
Andiswa
atomic model john dalton
Alliieyza Reply
Hlw,, I want to join.
crippah
what is a normal force
tumelo Reply
normal force is the line that is perpendicular to the surface
Thabo
normal force is the force exerted to an object by the surface , perpendicular to the surface
Fhatani
I want to join
Londiwe
better explanation of FARADAY'S LAW OF INDUCTION
Kgaugelo
what is the formular of energy
Wilka Reply
E=f\q
Wonder
E=hf
Mondli
doesnt it like depend on the type of energy
Benjamin
what does (1s2 2s2 and 2p6) mean?
Ryan Reply
That is Albau Diagram I guess!! it represent number of electrons in each energy orbit!! the numbers 1s, 2s&2p, 3s and so on represent energy orbit and then the other numbers after s & p represent number of electrons
Andiswa
That is SP notation
Kgaugelo
What are the differences between pd and emf?
Santosh Reply
How to calculate magnitude of Friction force
Tanaka Reply
in Newton
Obakeng
yeh
Obakeng
in newton
Fikile
Onother Question?
crippah
yes bruh
Fikile
First of all in wat Grade ae u..?
crippah
grde 11
Fikile
Oka nd hw is the properties of a substance afected by the intermoleculaer forces...
crippah
hi guys I m in grade 11 I need help with electrostatics
Serati
what is newton
Ndamulelo Reply
look for the x components n y components then after y add the x components separate n y separate then u use the Pythagoras theorem 2 find the resultand
Abigail Reply
Thank you very much
Luvuyo
how do I find 🔍 the critical angle
Ntandokazi
angle of incident and angle of reflection must always approach normal line thats where u will see 90° which is critical point
Tumelo
how to calculate the resultant net force
Luvuyo Reply

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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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