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4. Forces and Vectors

From Mechanics

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(New page: 4. Forces and Vectors Key Points <math>\begin{align} & \mathbf{F}=F\cos \alpha \mathbf{i}+F\cos (90-\alpha )\mathbf{j} \\ & =F\cos \alpha \mathbf{i}+F\sin \alpha \mathbf{j} \end{al...)
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4. Forces and Vectors
 
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Key Points
 
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<math>\begin{align}
<math>\begin{align}
& \mathbf{F}=F\cos \alpha \mathbf{i}+F\cos (90-\alpha )\mathbf{j} \\
& \mathbf{F}=F\cos \alpha \mathbf{i}+F\cos (90-\alpha )\mathbf{j} \\
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is called the vertical component of the force.
is called the vertical component of the force.
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'''[[Example 4.1]]'''
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Example 4.1
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Express each of the forces given below in the form ai + bj.
Express each of the forces given below in the form ai + bj.
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'''Solution'''
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Solution
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(a)
(a)
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Note the negative sign here in the first term.
Note the negative sign here in the first term.
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Example 4.2
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'''[[Example 4.2]]'''
Express the force shown below as a vector in terms of i and j.
Express the force shown below as a vector in terms of i and j.
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Solution
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'''Solution'''
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Note the negative sign in the second term.
Note the negative sign in the second term.
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Example 4.3
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'''[[Example 4.3]]'''
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Solution
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'''Solution'''
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Note that here both terms are negative.
Note that here both terms are negative.
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Example 4.4
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'''[[Example 4.4]]'''
Find the magnitude of the force (4i - 8j) N. Draw a diagram to show the direction of this force.
Find the magnitude of the force (4i - 8j) N. Draw a diagram to show the direction of this force.
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Solution
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'''Solution'''
The magnitude, FN , of the force is given by,
The magnitude, FN , of the force is given by,
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The angle, , is given by,
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The angle, <math>\theta </math>, is given by,
<math>\theta ={{\tan }^{-1}}\left( \frac{8}{4} \right)=63.4{}^\circ </math>
<math>\theta ={{\tan }^{-1}}\left( \frac{8}{4} \right)=63.4{}^\circ </math>
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Example 4.5
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'''[[Example 4.5]]'''
Find the magnitude and direction of the resultant of the four forces shown in the diagram.
Find the magnitude and direction of the resultant of the four forces shown in the diagram.
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Solution
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'''Solution'''
Force Vector Form
Force Vector Form
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The angle can be found using tan.
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The angle <math>\theta </math> can be found using tan.

Revision as of 15:48, 18 March 2009

F=Fcosi+Fcos(90−)j=Fcosi+Fsinj





Fcos is one component of the force. If i is horizontal, Fcos is called the horizontal component of the force.


Fsin is another component of the force. If j is vertical, Fsin is called the vertical component of the force.

Example 4.1

Express each of the forces given below in the form ai + bj.



Solution

(a)

20cos40i+20sin40j


(b)

−80cos30i+80sin30j


Note the negative sign here in the first term.

Example 4.2

Express the force shown below as a vector in terms of i and j.



Solution


28cos30i−28sin30j


Note the negative sign in the second term.

Example 4.3



Solution


−50cos44i−50sin44j


Note that here both terms are negative.

Example 4.4

Find the magnitude of the force (4i - 8j) N. Draw a diagram to show the direction of this force.

Solution

The magnitude, FN , of the force is given by,

F=42+82=80=894 N (to 3sf) 


The angle, , is given by,

=tan−148=634 


Example 4.5

Find the magnitude and direction of the resultant of the four forces shown in the diagram.



Solution

Force Vector Form 20 N 20cos50i+20sin50j

18 N −18j

25 N −25cos20i−25sin20j

15 N −15cos30i+15sin30j


Resultant Force = 20cos50−25cos20−15cos30i+20sin50−18−25sin20+15sin30j=−23627i−3730j


The magnitude is given by:


236272+37302=239 N (to 3sf) 


The angle can be found using tan.


tan=373023627=90

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