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Write force OA in component form.

Component of a vector

Component of a vector.

F x = 10 sin 30 0 = 10 X 1 2 = 5 N F y = 10 cos 30 0 = 10 X 3 2 = 5 3 N F = F x i F y j = 5 i 5 3 j N

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Write force OA in component form.

Component of a vector

Component of a vector.

F x = 10 sin 30 0 = 10 X 1 2 = 5 N F y = 10 cos 30 0 = 10 X 3 2 = 5 3 N F = F x i F y j = 5 i 5 3 j N

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Vector addition : algebraic method

Graphical method is meticulous and tedious as it involves drawing of vectors on a scale and measurement of angles. More importantly, it does not allow algebraic operations that otherwise would give a simple solution. We can, however, extend algebraic techniques to vectors, provided vectors are represented on a rectangular coordinate system . The representation of a vector on coordinate system uses the concept of unit vector and component.

Now, the stage is set to design a frame work, which allows vector addition with algebraic methods. The frame work for vector addition draws on two important concepts. The first concept is that a vector can be equivalently expressed in terms of three component vectors :

a = a x i + a y j + a z k b = b x i + b y j + b z k

The component vector form has important significance. It ensures that component vectors to be added are restricted to three known directions only. This paradigm eliminates the possibility of unknown direction. The second concept is that vectors along a direction can be treated algebraically. If two vectors are along the same line, then resultant is given as :

When θ = 0°, cosθ = cos 0° = 1 and,

R = ( P 2 + 2 P Q + Q 2 ) = { ( P + Q ) 2 } = P + Q

When θ = 180°, cosθ = cos 180° = -1 and,

R = ( P 2 - 2 P Q + Q 2 ) = { ( P - Q ) 2 } = P - Q

Thus, we see that the magnitude of the resultant is equal to algebraic sum of the magnitudes of the two vectors.

Using these two concepts, the addition of vectors is affected as outlined here :

1: Represent the vectors ( a and b ) to be added in terms of components :

a = a x i + a y j + a z k b = b x i + b y j + b z k

2: Group components in a given direction :

a + b = a x i + a y j + a z k + b x i + b y j + b z k a + b = ( a x + b x ) i + ( a y + b y ) j + ( a z + b z ) k

3: Find the magnitude and direction of the sum, using analytical method

a = { ( a x + b x ) 2 + ( a y + b y ) 2 + ( a z + b z ) 2 )

Exercises

Find the angle that vector √3 i - j makes with y-axis.

From graphical representation, the angle that vector makes with y-axis has following trigonometric function :

Angle

Vector makes an angle with y-axis.

tan θ = a x a y

Now, we apply the formulae to find the angle, say θ, with y-axis,

tan θ = - 3 1 = - 3 = tan 120 °

θ = 120 °

In case, we are only interested to know the magnitude of angle between vector and y-axis, then we can neglect the negative sign,

tan θ’ = 3 = tan 60 °

θ’ = 60 °

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If a vector makes angles α,β and γ with x,y and z axes of a rectangular coordinate system, then prove that :

cos 2 α + cos 2 β + cos 2 γ = 1

The vector can be expressed in terms of its component as :

a = a x i + a y j + a z k

where a x = a cos α ; a y = a cos β ; a z = a cos γ .

Angles

Vector makes different angles with axes..

The magnitude of the vector is given by :

a = ( a x 2 + a y 2 + a z 2 )

Putting expressions of components in the equation,

a = ( a 2 cos 2 α + a 2 cos 2 β + a 2 cos 2 γ )

( cos 2 α + cos 2 β + cos 2 γ ) = 1

Squaring both sides,

cos 2 α + cos 2 β + cos 2 γ = 1

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Find the components of weight of a block along the incline and perpendicular to the incline.

The component of the weight along the incline is :

Weight on an incline

Components of weight along the incline and perpendicualt to the incline.

W x = mg sin θ = 100 x sin 30 ° = 100 x 1 2 = 50 N

and the component of weight perpendicular to incline is :

W y = mg cos θ = 100 x cos 30 ° = 100 x 3 2 = 50 3 N

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The sum of magnitudes of two forces acting at a point is 16 N. If the resultant of the two forces is 8 N and it is normal to the smaller of the two forces, then find the forces.

We depict the situation as shown in the figure. The resultant force R is shown normal to small force F 1 . In order that the sum of the forces is equal to R, the component of larger force along the x-direction should be equal to smaller force :

Two forces acting on a point

The resultant force is perpendicular to smaller of the two forces.

F 2 sin θ = F 1

Also, the component of the larger force along y-direction should be equal to the magnitude of resultant,

F 2 cos θ = R = 8

Squaring and adding two equations, we have :

F 2 2 = F 1 2 + 64

F 2 2 - F 1 2 = 64

However, according to the question,

F 1 + F 2 = 16

Substituting, we have :

F 2 - F 1 = 64 16 = 4

Solving,

F 1 = 6 N F 2 = 10 N

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More illustrations on the subject are available in the module titled Resolution of forces

Questions & Answers

what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
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Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
AMJAD
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
AMJAD
what is system testing
AMJAD
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
can nanotechnology change the direction of the face of the world
Prasenjit Reply
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
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the Beer law works very well for dilute solutions but fails for very high concentrations. why?
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how did you get the value of 2000N.What calculations are needed to arrive at it
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can anyone tell who founded equations of motion !?
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n=a+b/T² find the linear express
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Moment of inertia of a bar in terms of perpendicular axis theorem
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Source:  OpenStax, Physics for k-12. OpenStax CNX. Sep 07, 2009 Download for free at http://cnx.org/content/col10322/1.175
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