4. Forces and Vectors - Revision history

Ian at 15:01, 19 March 2009

Ian — Thu, 19 Mar 2009 15:01:31 GMT

←Older revision		Revision as of 15:01, 19 March 2009
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	'''Solution'''		'''Solution'''
		+
		+	[[Image:TF4.4.GIF]]
		+

	The magnitude, FN , of the force is given by,		The magnitude, FN , of the force is given by,
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		+	[[Image:TF4.5.GIF]]


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	& \theta =9.0{}^\circ		& \theta =9.0{}^\circ
	\end{align}</math>		\end{align}</math>
		+
		+
		+	[[Image:TF4.5a.GIF]]

Ian at 14:50, 19 March 2009

Ian — Thu, 19 Mar 2009 14:50:32 GMT

←Older revision		Revision as of 14:50, 19 March 2009
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-		+	[[Image:TF.teori.GIF]]


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	<math>F\cos \alpha </math>		<math>F\cos \alpha </math>
-	is one component of the force. If i is horizontal,	+	is one component of the force. If <math>\mathbf{i}</math> is horizontal,
	<math>F\cos \alpha </math>		<math>F\cos \alpha </math>
	is called the horizontal component of the force.		is called the horizontal component of the force.
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	<math>F\sin \alpha </math>		<math>F\sin \alpha </math>
-	is another component of the force. If j is vertical,	+	is another component of the force. If <math>\mathbf{j}</math> is vertical,
	<math>F\sin \alpha </math>		<math>F\sin \alpha </math>
	is called the vertical component of the force.		is called the vertical component of the force.
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	'''[[Example 4.1]]'''		'''[[Example 4.1]]'''

-	Express each of the forces given below in the form ai + bj.	+	Express each of the forces given below in the form a<math>\mathbf{i}</math> + b<math>\mathbf{j}</math>.

		+	(a)

		+	[[Image:TF4.1a.GIF]]

		+	(b)

		+	[[Image:TF4.1b.GIF]]

	'''Solution'''		'''Solution'''
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	Note the negative sign here in the first term.		Note the negative sign here in the first term.
		+
		+

	'''[[Example 4.2]]'''		'''[[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 <math>\mathbf{i}</math> and <math>\mathbf{j}</math>.
-		+


		+	[[Image:TF4.2.GIF]]


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	Note the negative sign in the second term.		Note the negative sign in the second term.
		+
		+

	'''[[Example 4.3]]'''		'''[[Example 4.3]]'''


		+	Express the force shown below as a vector in terms of
		+	<math>\mathbf{i}</math>
		+	and
		+	<math>\mathbf{j}</math>

		+
		+	[[Image:TF4.3.GIF]]


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	Note that here both terms are negative.		Note that here both terms are negative.
		+
		+

	'''[[Example 4.4]]'''		'''[[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 (4<math>\mathbf{i}</math> - 8<math>\mathbf{j}</math>) N. Draw a diagram to show the direction of this force.

	'''Solution'''		'''Solution'''

Ian at 15:48, 18 March 2009

Ian — Wed, 18 Mar 2009 15:48:56 GMT

←Older revision		Revision as of 15:48, 18 March 2009
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-	4. Forces and Vectors
-
-	Key Points
-
-
-
-
	<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.

-		+	'''[[Example 4.1]]'''
-		+
-	Example 4.1	+

	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'''
-	Solution	+

	(a)		(a)
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	Note the negative sign here in the first term.		Note the negative sign here in the first term.

-	Example 4.2	+	'''[[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	+	'''Solution'''


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	Note the negative sign in the second term.		Note the negative sign in the second term.

-	Example 4.3	+	'''[[Example 4.3]]'''


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-	Solution	+	'''Solution'''


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	Note that here both terms are negative.		Note that here both terms are negative.

-	Example 4.4	+	'''[[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.

-	Solution	+	'''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,	+	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>


-	Example 4.5	+	'''[[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	+	'''Solution'''

	Force Vector Form		Force Vector Form
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-	The angle  can be found using tan.	+	The angle <math>\theta </math> can be found using tan.

Ian: New page: 4. Forces and Vectors Key Points $\begin{align} & \mathbf{F}=F\cos \alpha \mathbf{i}+F\cos (90-\alpha )\mathbf{j} \\ & =F\cos \alpha \mathbf{i}+F\sin \alpha \mathbf{j} \end{al...$

Ian — Wed, 18 Mar 2009 15:32:25 GMT

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...

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{align}</math>

<math>F\cos \alpha </math>
is one component of the force. If i is horizontal,
<math>F\cos \alpha </math>
is called the horizontal component of the force.

<math>F\sin \alpha </math>
is another component of the force. If j is vertical,
<math>F\sin \alpha </math>
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)

<math>20\cos 40{}^\circ \mathbf{i}+20\sin 40{}^\circ \mathbf{j}</math>

(b)

<math>-80\cos 30{}^\circ \mathbf{i}+80\sin 30{}^\circ \mathbf{j}</math>

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

<math>28\cos 30{}^\circ \mathbf{i}-28\sin 30{}^\circ \mathbf{j}</math>

Note the negative sign in the second term.

Example 4.3

Solution

<math>-50\cos 44{}^\circ \mathbf{i}-50\sin 44{}^\circ \mathbf{j}</math>

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,

<math>F=\sqrt{{{4}^{2}}+{{8}^{2}}}=\sqrt{80}=8.94\text{ N (to 3sf)}</math>

The angle, , is given by,

<math>\theta ={{\tan }^{-1}}\left( \frac{8}{4} \right)=63.4{}^\circ </math>

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
<math>20\cos 50{}^\circ \mathbf{i}+20\sin 50{}^\circ \mathbf{j}</math>

18 N
<math>-18\mathbf{j}</math>

25 N
<math>-25\cos 20{}^\circ \mathbf{i}-25\sin 20{}^\circ \mathbf{j}</math>

15 N
<math>-15\cos 30{}^\circ \mathbf{i}+15\sin 30{}^\circ \mathbf{j}</math>

<math>\begin{align}
& \text{Resultant Force }=\text{ }\left( 20\cos 50{}^\circ -25\cos 20{}^\circ -15\cos 30{}^\circ \right)\mathbf{i}+\left( 20\sin 50{}^\circ -18-25\sin 20{}^\circ +15\sin 30{}^\circ \right)\mathbf{j} \\
& =-23.627\mathbf{i}-3.730\mathbf{j}
\end{align}</math>

The magnitude is given by:

<math>\sqrt{{{23.627}^{2}}+{{3.730}^{2}}}=23.9\text{ N (to 3sf)}</math>

The angle  can be found using tan.

<math>\begin{align}
& \tan \theta =\frac{3.730}{23.627} \\
& \theta =9.0{}^\circ
\end{align}</math>

4. Forces and Vectors - Revision history

Ian at 15:01, 19 March 2009

Ian at 14:50, 19 March 2009

Ian at 15:48, 18 March 2009

Ian: New page: 4. Forces and Vectors Key Points \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...

Ian: New page: 4. Forces and Vectors Key Points $\begin{align} & \mathbf{F}=F\cos \alpha \mathbf{i}+F\cos (90-\alpha )\mathbf{j} \\ & =F\cos \alpha \mathbf{i}+F\sin \alpha \mathbf{j} \end{al...$