Components Of Vectors

Vectors in physics. Vectors and its components. Splitting of vectors into its components. Found of the components of vectors. Addition of the components of the vectors.


1- Splitting A Vector Into Its Components

To get the vertical and horizontal components of the vectors, the vector are separates into two components that it’s perpendicular to each other using coordinate system.

If the angle between the x axis with the vector is "α", the components of the vector is found as the following.

Vectorcompond-s1i1


Cosα =Fx
F




Fx = F•cosα

Sinα =Fy
F




Fy = F•sinα


Fx: The horizontal component.

Fy: The vertical component.


Example:

Vectorcompond-s1i2


The magnitude of the vector F1 that seen in the figure is 80 N, the angle between with x axis and its is 53 degrees.

The magnitude of the vector F2 is 50 N, and the angle between with x axis and its is 53 degrees.

What is the resultant of the F1 + F2


(sin53° = 0.8, cos53°=0.6, cos37°=0.8, sin37°=0.6)


Solution:

We separates the vectors into its components and add the components.

First, let we separate the vectors into its components.

The components of the vector F1.

Cosα =Fx
F1




F1x = F1•cos53°

F1x = 80•cos53°

F1x = 80•0.6

F1x = 48 N


Sin53 =F1y
F1




F1y = F1•sin53°

F1y = 80•sin53°

F1y = 80•0.8

F1y = 64 N


Vectorcompond-s1i3


The components of the vector F2.

The vector F2 is doing angle of 53 degrees from left side with x axis. The angle that vector F2 doing with x axis is 180 + 53 = 233 degrees.


Cos233° = Cos(180 + 53) = -Cos53° 

F2x = - 50•0.6

F2x = - 30 N


Sin233° = Sin(180 + 53) = -Sin53° 

F2y =- 50•0.8

F2y = -40 N


Vectorcompond-s1i4



Now, let we add, the components with the same direction.

Ftx = F1x + F2x

Ftx = 48 – 30

Ftx = 18 N


Fty = F1y + F2y

Fty = 64 - 40

Fty = 24 N


Now, we have two vectors that are perpendicular to each other.

The resultant vector can be found using pythagorean theorem.

Vectorcompond-s1i5



Ft2 = 182 + 242

(Ft)2  = 324 + 576

(Ft)2 = 900

Ft = 30 N


If we want find the angle that vector Ft does with x axis, we can use equality in the following.

tanα = 24
18




tanα = 1.33

α = tan-1 1.33

α = 53 degrees.



Example:

Vectorcompond-s1i6


Find the resultant of three forces that affected to the object K.

 (sin53° = 0.8, cos53°=0.6, cos37°=0.8, sin37°=0.6)


Solution:

First we must separate the forces into its components.

Next, we add the components that have same direction among themselves.

So, we add the components that on the x axis among themselves and we add the components that on the y axis among themselves too.


Vectorcompond-s1i7



F1x = 200•cos( – 37)

F1x = 200•0.8

F1x = 160 N


F1y = 200•sin(- 37)

F1y = -200•0.6


F1y = - 120 N

Vectorcompond-s1i8


F2x = 50•cos37°

F2x = 50•0.8

F2x = 40 N


F2y = 50•sin37°

F2y = 50•0.6

F2y = 30 N


Vectorcompond-s1i9



F3x = 100•cos(-37)

F3x = 100•0.8

F3x = 80 N


F3y = 100•sin(-37)

F3y = -100•0.6

F3y = -60 N



Let we add the components among themselves.

Fx = F1x + F2x + F3x

Fx = 160 + 40 + 80

Fx = 280 N


Fy = F1y + F2y + F3y

F2 = -120 + 30 – 60

F2 = - 150 N

Finally, we have two forces perpendicular to each other. We can find the resultant of these vectors using pythagorean theorem.


Vectorcompond-s1i10



F = 1502 + 2802

F = 22500 + 78400

F = 100900

F = 317.65 N



Subject Expression of Vectors



RISE KNOWLEDGE

14/09/2018

  • WRITE COMMENT
  • NAME SURNAME(or nick)
  • COMMENT
COPYRIGHT© ALL RIGHTS RESERVED