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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 resistance in grade 11
Ngomane Reply
last year memo in 2018 June exam
Ngomane
states the Newton's second law of motion
Shallin Reply
In words it says:" when a net force is applied to an object of mass, It accelerates in the direction of the net force. The acceleration is directly proportional to the net force and inversely proportional to the mass".
Buhle
And in symbols: Fnet = ma
Buhle
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Shallin
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Buhle
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Ngomane
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Mzondi
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manu
which degree is usually be a refractive angle before it complete a totally reflection?
Mzondi
90°
Boss
yes obviously 90 degree
manu
how does hydrogen bonds differ from London force
Madzivha Reply
how come the resultant force is 0
Andrew Reply
It's when you have equivalent forces going different directions then your resultant will be equal to zero
Temosho
describe what john's experiment proves about water molecules?
Fanozi Reply
Newton's first law of motion
Ayabonga Reply
hy
Madzivha
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Ayah
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Dire
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Precious
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Precious
great
Dire
An unknown gas has pressure,volume and temprature of 0.9atm,and 120°C.how many moles of gas are present?
Chrislyn Reply
Can you, if possible send me more quizzes
Bradley Reply
What is selmon
heath Reply
how long it takes for 25ml ethanol to be evaporated?
Kgaugelo Reply
how to calculate acceleration
Sphe Reply
It depends
Mbongeni
Please state the Newton third low
Malwandla
hy
Nelito
Newton's Third law states that to every force applied, there's an equal but opposite reaction
Lala
difference between a head and tail methods
Zenande Reply
difference between a head-to-tail and tail-to-tail
Zenande
head to tail you draw each vector starting head of the previous vector and tail to tail you construct a parallelogram whereas you started the two vectors from the same axis(from their tails)and the diagonal between the vectors is the resultant vector,tail to tail only includes two vectors
Mbongeni
what is a normal force
Zenande Reply
a normal force is the force that the surface applies on the object. The force is perpendicular to the surface.
Minaa
Lewis structure for C2H2
Mthokozisi Reply
H-C-H
Koketso
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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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