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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

what is reduction
Lesego Reply
reduction is the gain of electrons of an atom ; compound or molecule. reduction is also the decrease of the oxidation number of an atom ; compound / molecule.
What does Redox reactins mean?
kholeka Reply
Reaction in which there is transfer of electrons between elements
how to calculate a vector
sakhumzi Reply
state Faraday's law in word
Emkay Reply
which rules do you have to follow to identify oxidation numbers
sessy Reply
there are 7 of them > look them up on siyavula textbook physics
Where can I get a Siyavula textbook?
if possibly ther was a way to send a document I told hv sent it to you
go to siyavula on Google where u can download the textbook
Why do vectors have arrows?
vanessa Reply
vector quantity has both direction and magnitude so arrows illustrate the direction of the vector
how to calculate net electric field at a point
Bazel Reply
what are controlled variables
Link Reply
Independent variables in an experiment which influence the outcome results
Such as temperature when dealing with ohms law
variables whose amount is determined by the person doing the experiment
how to calculate exodation number
Cphiwe Reply
Calculate relative atomic mass
Lindo Reply
what is chemical bonding?
Lefa Reply
Chemical Bonding is a mutual attraction between two atoms resulting from the simultaneous attraction between their nuclei and the outer elections
How would you go about finding the resistance of an unknown resistor using only power supply, a voltmeter and a known resistance
who is the scientist who discovered electromagnetism
Zivele Reply
what happens to the galvanometer when the switch is closed ?
how do we identify an oxidising or reducing agent in a reaction?
@zivele in your products the element with an oxidation number less than 0 is a reducing and the one with an oxidation number greater than 0 is oxidising
what is electricity
Vihanga Reply
Formula for concentration
Kabelo Reply
if given number of moles and volume , can use c=n/V
Chemistry term three topic is stressing me out, I really need a huge help
on what
during a snooker competition ,a 200g ball A m moving with velocity va collide head on with a identical ball B that was at rest.A after the collision ball A remains at rest wile ball B moves on with a velocity of 4m/s? With what speed was ball a moving before the collision
mathew 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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