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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

explain why HCL is a Bronstead lowry acid . use an equation of HCL in water to explain
Mihlali Reply
what is electron. ( you should not allowed to tell it properties in it defination.)
vigyan Reply
investigative question
Obakeng Reply
how do you create a bomb
Nasty Reply
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Nasty
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Nasty
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Nasty
if you want to create bomb use this chemical substance oxidised, sulphuric acid,tobacco,gas and flammable substance , mix all this substance together and put in strong mental bottle like gas bottle put in . you must close the bottle very parked and carry that bottle carefully and shake in 5 minutes
Aaron
after shake put this bottle were the temperature is 100 degrees Celsius after moment that bottle will be high explosion like bomb
Aaron
what is mass
asmerom Reply
is object amount matter contain by object
Aaron
what is hypothesis
Aaron
hypothesis is a tentative explanation that does not have facts and it can be proven to be wrong or right
MPHO
wath is physical science
Siphelo Reply
what is physics
Dineo Reply
the branch of science concerned with the nature and properties of matter and energy
Thabisto
use a vector scale diagram and represent the two displacements(5m east and 7m west)
itumeleng Reply
what os covalent?
Shellah Reply
can i ask something?
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Precious
how to calculate enthalpy on a reaction?
Uapfa
how do you learn this chapter festet
Lisakhanya Reply
it's not all about learning it fast but understanding the basics, the concepts. formulas .. if you juss understand the basics you know the chapter .understand it
Precious
you are right precious
MPHO
can i ask smething.?
Uapfa
electric field around a point
Sameer Reply
The force per positive unit charge
Mpho
what is Kevlar made of ?
Crazy Reply
How do I determine the number of valence electrons for an atom
Kamohelo Reply
how to find a point between to charges that is zero?
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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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