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scaling.html
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scaling.html
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<!DOCTYPE HTML>
<html lang="en">
<head>
<title> Quiz 1: Estimation models - Scaling laws </title>
<meta http-equiv="Content-Type" content="text/html; charset=utf-8">
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<meta name="authors" content="WebQuiz: Andrew Mathas">
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<meta name="Copyright" content="WebQuiz: 2004-2019 Andrew Mathas and Donald Taylor, University of Sydney">
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<table class="marking-key">
<tr><td class="star">★</td><td style="width: 14ex;">right first attempt</td></tr>
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</div>
</div>
<div class="quiz-questions">
<div class="quiz-header">
<div class="quiz-title">Quiz 1: Estimation models - Scaling laws</div><div></div>
<span id="question-label" class="question-label">Question</span>
<span id="question-number" class="question-label">Recall on scaling laws</span>
<span class="arrows">
<a onClick="nextQuestion(-1);" title="Previous unanswered question">◄</a>
<span class="question-label">Questions</span>
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<div id="question-1" class="question" style="display:none;">
<br
class="newline" /><br
class="newline" />Assumptions : <div class="minipage"> <ul class="itemize1">
<li class="itemize">Geometrical similarity
</li>
<li class="itemize">Material similarity
</li>
<li class="itemize">One dominant phenomena</li></ul>
</div>
<p>Mathematical form: <!--l. 26--><math
xmlns="http://www.w3.org/1998/Math/MathML"
display="inline" ><mi
>y</mi> <mo
class="MathClass-rel">=</mo> <mi
>k</mi><msup><mrow
><mi
>L</mi></mrow><mrow
><mi
>a</mi></mrow></msup
></math>
<br
class="newline" />with k function of reference and a of physical effect
</p>
<p>Notation: <!--l. 29--><math
xmlns="http://www.w3.org/1998/Math/MathML"
display="inline" ><msup><mrow
><mi
>y</mi></mrow><mrow
><mo
class="MathClass-bin">∗</mo></mrow></msup
> <mo
class="MathClass-rel">=</mo> <msup><mrow
><mi
>L</mi></mrow><mrow
><mo
class="MathClass-bin">∗</mo></mrow></msup
><mi
>a</mi></math>
</p>
<p>Obtention ways: </p>
<div class="minipage"> <ul class="itemize1">
<li class="itemize">direct manipulation of equations
</li>
<li class="itemize">dimensional analysis and Buckingham theorem
</li>
<li class="itemize">One dominant phenomena</li></ul>
</div> Components: One main design driver express by a constant stress X*=1
<div class="center"
>
<p>
</p>
<p><img
src="scaling/Picture1.png" alt="PIC" width="339" height="199" /></p>
</div>
</div>
<div id="question1" class="question" style="display:none;">
<span class="question-label"></span>
<div class="center"
>
Geometric similarity</div>
We assume to have similarity on all geometrical parameters :
<!--l. 50--><math
xmlns="http://www.w3.org/1998/Math/MathML"
display="inline" ><msup><mrow
><mi
>r</mi></mrow><mrow
><mo
class="MathClass-bin">∗</mo></mrow></msup
> <mo
class="MathClass-rel">=</mo> <msup><mrow
><mi
>d</mi></mrow><mrow
><mo
class="MathClass-bin">∗</mo></mrow></msup
> <mo
class="MathClass-rel">=</mo> <mi
>…</mi> <mo
class="MathClass-rel">=</mo> <msup><mrow
><mi
>l</mi></mrow><mrow
><mo
class="MathClass-bin">∗</mo></mrow></msup
></math>
<br
class="newline" /><br
class="newline" />Give evolutions of areas :
<form id="Q1Form" onSubmit="return false;">
<table class="question-choices"><tr><td><input type="radio" name="Q1option"/></td>
<td class="brown" >(a)</td><td><div class="question-choices"><!--l. 54--><math
xmlns="http://www.w3.org/1998/Math/MathML"
display="inline" ><msup><mrow
><mi
>l</mi></mrow><mrow
><mo
class="MathClass-bin">∗</mo></mrow></msup
></math>
<a
id="x1-5r2"></a></div></td>
</tr><!-- part=0, cols=1, # answers = 4 -->
<tr><td><input type="radio" name="Q1option"/></td>
<td class="brown" >(b)</td><td><div class="question-choices"><!--l. 55--><math
xmlns="http://www.w3.org/1998/Math/MathML"
display="inline" ><msup><mrow
><mi
>l</mi></mrow><mrow
><msup><mrow
><mo
class="MathClass-bin">∗</mo></mrow><mrow
><mn>2</mn></mrow></msup
>
</mrow></msup
></math>
<a
id="x1-6r3"></a></div></td>
</tr><!-- part=1, cols=1, # answers = 4 -->
<tr><td><input type="radio" name="Q1option"/></td>
<td class="brown" >(c)</td><td><div class="question-choices"><!--l. 56--><math
xmlns="http://www.w3.org/1998/Math/MathML"
display="inline" ><msup><mrow
><mi
>l</mi></mrow><mrow
><msup><mrow
><mo
class="MathClass-bin">∗</mo></mrow><mrow
><mo
class="MathClass-bin">−</mo><mn>2</mn></mrow></msup
>
</mrow></msup
></math>
<a
id="x1-7r4"></a></div></td>
</tr><!-- part=2, cols=1, # answers = 4 -->
<tr><td><input type="radio" name="Q1option"/></td>
<td class="brown" >(d)</td><td><div class="question-choices"><!--l. 57--><math
xmlns="http://www.w3.org/1998/Math/MathML"
display="inline" ><msup><mrow
><mi
>l</mi></mrow><mrow
><msup><mrow
><mo
class="MathClass-bin">∗</mo></mrow><mrow
><mfrac><mrow
><mn>1</mn></mrow>
<mrow
><mn>2</mn></mrow></mfrac> </mrow></msup
>
</mrow></msup
></math></div></td>
</tr><!-- part=3, cols=1, # answers = 4 -->
</table>
<p>
<input type="button" value="Check answer" name="answer" class="input-button" onClick="checkAnswer(1);"/>
<input type="button" value="Next unanswered question" class="input-button" title="Next unanswered question" name="next" onClick="nextQuestion(1);"/>
</p>
</form>
<div class="answer">
<div id="q1feedback1" class="feedback">
<em>Choice (a) is <span class="dazzle">Incorrect.</span></em>
<div></div>
</div>
<div id="q1feedback2" class="feedback">
<em>Choice (b) is <span class="dazzle">Incorrect.</span></em>
<div></div>
</div>
<div id="q1feedback3" class="feedback">
<em>Choice (c) is <span class="dazzle">Incorrect.</span></em>
<div></div>
</div>
<div id="q1feedback4" class="feedback">
<em>Choice (d) is <span class="dazzle">Correct!</span></em>
<div></div>
</div></div>
</div>
<div id="question2" class="question" style="display:none;">
<span class="question-label"></span>
<div class="center"
>
Geometric similarity</div>
We assume to have similarity on all geometrical parameters :
<!--l. 65--><math
xmlns="http://www.w3.org/1998/Math/MathML"
display="inline" ><msup><mrow
><mi
>r</mi></mrow><mrow
><mo
class="MathClass-bin">∗</mo></mrow></msup
> <mo
class="MathClass-rel">=</mo> <msup><mrow
><mi
>d</mi></mrow><mrow
><mo
class="MathClass-bin">∗</mo></mrow></msup
> <mo
class="MathClass-rel">=</mo> <mi
>…</mi> <mo
class="MathClass-rel">=</mo> <msup><mrow
><mi
>l</mi></mrow><mrow
><mo
class="MathClass-bin">∗</mo></mrow></msup
></math>
<br
class="newline" /><br
class="newline" />Give evolutions of volumes :
<form id="Q2Form" onSubmit="return false;">
<table class="question-choices"><tr><td><input type="radio" name="Q2option"/></td>
<td class="brown" >(a)</td><td><div class="question-choices"><!--l. 69--><math
xmlns="http://www.w3.org/1998/Math/MathML"
display="inline" ><msup><mrow
><mi
>l</mi></mrow><mrow
><mo
class="MathClass-bin">∗</mo></mrow></msup
></math>
<a
id="x1-10r2"></a></div></td>
</tr><!-- part=0, cols=1, # answers = 5 -->
<tr><td><input type="radio" name="Q2option"/></td>
<td class="brown" >(b)</td><td><div class="question-choices"><!--l. 70--><math
xmlns="http://www.w3.org/1998/Math/MathML"
display="inline" ><msup><mrow
><mi
>l</mi></mrow><mrow
><msup><mrow
><mo
class="MathClass-bin">∗</mo></mrow><mrow
><mn>2</mn></mrow></msup
>
</mrow></msup
></math>
<a
id="x1-11r3"></a></div></td>
</tr><!-- part=1, cols=1, # answers = 5 -->
<tr><td><input type="radio" name="Q2option"/></td>
<td class="brown" >(c)</td><td><div class="question-choices"><!--l. 71--><math
xmlns="http://www.w3.org/1998/Math/MathML"
display="inline" ><msup><mrow
><mi
>l</mi></mrow><mrow
><msup><mrow
><mo
class="MathClass-bin">∗</mo></mrow><mrow
><mn>3</mn></mrow></msup
>
</mrow></msup
></math>
<a
id="x1-12r4"></a></div></td>
</tr><!-- part=2, cols=1, # answers = 5 -->
<tr><td><input type="radio" name="Q2option"/></td>
<td class="brown" >(d)</td><td><div class="question-choices"><!--l. 72--><math
xmlns="http://www.w3.org/1998/Math/MathML"
display="inline" ><msup><mrow
><mi
>l</mi></mrow><mrow
><msup><mrow
><mo
class="MathClass-bin">∗</mo></mrow><mrow
><mo
class="MathClass-bin">−</mo><mn>3</mn></mrow></msup
>
</mrow></msup
></math>
<a
id="x1-13r5"></a></div></td>
</tr><!-- part=3, cols=1, # answers = 5 -->
<tr><td><input type="radio" name="Q2option"/></td>
<td class="brown" >(e)</td><td><div class="question-choices"><!--l. 73--><math
xmlns="http://www.w3.org/1998/Math/MathML"
display="inline" ><msup><mrow
><mi
>l</mi></mrow><mrow
><msup><mrow
><mo
class="MathClass-bin">∗</mo></mrow><mrow
><mfrac><mrow
><mn>1</mn></mrow>
<mrow
><mn>3</mn></mrow></mfrac> </mrow></msup
>
</mrow></msup
></math></div></td>
</tr><!-- part=4, cols=1, # answers = 5 -->
</table>
<p>
<input type="button" value="Check answer" name="answer" class="input-button" onClick="checkAnswer(2);"/>
<input type="button" value="Next unanswered question" class="input-button" title="Next unanswered question" name="next" onClick="nextQuestion(1);"/>
</p>
</form>
<div class="answer">
<div id="q2feedback1" class="feedback">
<em>Choice (a) is <span class="dazzle">Incorrect.</span></em>
<div></div>
</div>
<div id="q2feedback2" class="feedback">
<em>Choice (b) is <span class="dazzle">Incorrect.</span></em>
<div></div>
</div>
<div id="q2feedback3" class="feedback">
<em>Choice (c) is <span class="dazzle">Incorrect.</span></em>
<div></div>
</div>
<div id="q2feedback4" class="feedback">
<em>Choice (d) is <span class="dazzle">Incorrect.</span></em>
<div></div>
</div>
<div id="q2feedback5" class="feedback">
<em>Choice (e) is <span class="dazzle">Correct!</span></em>
<div></div>
</div></div>
</div>
<div id="question3" class="question" style="display:none;">
<span class="question-label"></span>
<div class="center"
>
Geometric similarity</div>
We assume to have similarity on all geometrical parameters :
<!--l. 81--><math
xmlns="http://www.w3.org/1998/Math/MathML"
display="inline" ><msup><mrow
><mi
>r</mi></mrow><mrow
><mo
class="MathClass-bin">∗</mo></mrow></msup
> <mo
class="MathClass-rel">=</mo> <msup><mrow
><mi
>d</mi></mrow><mrow
><mo
class="MathClass-bin">∗</mo></mrow></msup
> <mo
class="MathClass-rel">=</mo> <mi
>…</mi> <mo
class="MathClass-rel">=</mo> <msup><mrow
><mi
>l</mi></mrow><mrow
><mo
class="MathClass-bin">∗</mo></mrow></msup
></math>
<br
class="newline" /><br
class="newline" />Give evolutions of masses :
<form id="Q3Form" onSubmit="return false;">
<table class="question-choices"><tr><td><input type="radio" name="Q3option"/></td>
<td class="brown" >(a)</td><td><div class="question-choices"><!--l. 85--><math
xmlns="http://www.w3.org/1998/Math/MathML"
display="inline" ><msup><mrow
><mi
>l</mi></mrow><mrow
><mo
class="MathClass-bin">∗</mo></mrow></msup
></math>
<a
id="x1-16r2"></a></div></td>
</tr><!-- part=0, cols=1, # answers = 5 -->
<tr><td><input type="radio" name="Q3option"/></td>
<td class="brown" >(b)</td><td><div class="question-choices"><!--l. 86--><math
xmlns="http://www.w3.org/1998/Math/MathML"
display="inline" ><msup><mrow
><mi
>l</mi></mrow><mrow
><msup><mrow
><mo
class="MathClass-bin">∗</mo></mrow><mrow
><mn>2</mn></mrow></msup
>
</mrow></msup
></math>
<a
id="x1-17r3"></a></div></td>
</tr><!-- part=1, cols=1, # answers = 5 -->
<tr><td><input type="radio" name="Q3option"/></td>
<td class="brown" >(c)</td><td><div class="question-choices"><!--l. 87--><math
xmlns="http://www.w3.org/1998/Math/MathML"
display="inline" ><msup><mrow
><mi
>l</mi></mrow><mrow
><msup><mrow
><mo
class="MathClass-bin">∗</mo></mrow><mrow
><mn>3</mn></mrow></msup
>
</mrow></msup
></math>
<a
id="x1-18r4"></a></div></td>
</tr><!-- part=2, cols=1, # answers = 5 -->
<tr><td><input type="radio" name="Q3option"/></td>
<td class="brown" >(d)</td><td><div class="question-choices"><!--l. 88--><math
xmlns="http://www.w3.org/1998/Math/MathML"
display="inline" ><msup><mrow
><mi
>l</mi></mrow><mrow
><msup><mrow
><mo
class="MathClass-bin">∗</mo></mrow><mrow
><mo
class="MathClass-bin">−</mo><mn>3</mn></mrow></msup
>
</mrow></msup
></math>
<a
id="x1-19r5"></a></div></td>
</tr><!-- part=3, cols=1, # answers = 5 -->
<tr><td><input type="radio" name="Q3option"/></td>
<td class="brown" >(e)</td><td><div class="question-choices"><!--l. 89--><math
xmlns="http://www.w3.org/1998/Math/MathML"
display="inline" ><msup><mrow
><mi
>l</mi></mrow><mrow
><msup><mrow
><mo
class="MathClass-bin">∗</mo></mrow><mrow
><mfrac><mrow
><mn>1</mn></mrow>
<mrow
><mn>3</mn></mrow></mfrac> </mrow></msup
>
</mrow></msup
></math></div></td>
</tr><!-- part=4, cols=1, # answers = 5 -->
</table>
<p>
<input type="button" value="Check answer" name="answer" class="input-button" onClick="checkAnswer(3);"/>
<input type="button" value="Next unanswered question" class="input-button" title="Next unanswered question" name="next" onClick="nextQuestion(1);"/>
</p>
</form>
<div class="answer">
<div id="q3feedback1" class="feedback">
<em>Choice (a) is <span class="dazzle">Incorrect.</span></em>
<div></div>
</div>
<div id="q3feedback2" class="feedback">
<em>Choice (b) is <span class="dazzle">Incorrect.</span></em>
<div></div>
</div>
<div id="q3feedback3" class="feedback">
<em>Choice (c) is <span class="dazzle">Incorrect.</span></em>
<div></div>
</div>
<div id="q3feedback4" class="feedback">
<em>Choice (d) is <span class="dazzle">Incorrect.</span></em>
<div></div>
</div>
<div id="q3feedback5" class="feedback">
<em>Choice (e) is <span class="dazzle">Correct!</span></em>
<div></div>
</div></div>
</div>
<div id="question4" class="question" style="display:none;">
<span class="question-label"></span>
<div class="center"
>
Geometric similarity</div>
We assume to have similarity on all geometrical parameters :
<!--l. 96--><math
xmlns="http://www.w3.org/1998/Math/MathML"
display="inline" ><msup><mrow
><mi
>r</mi></mrow><mrow
><mo
class="MathClass-bin">∗</mo></mrow></msup
> <mo
class="MathClass-rel">=</mo> <msup><mrow
><mi
>d</mi></mrow><mrow
><mo
class="MathClass-bin">∗</mo></mrow></msup
> <mo
class="MathClass-rel">=</mo> <mi
>…</mi> <mo
class="MathClass-rel">=</mo> <msup><mrow
><mi
>l</mi></mrow><mrow
><mo
class="MathClass-bin">∗</mo></mrow></msup
></math>
<br
class="newline" /><br
class="newline" />Give evolutions of intertias :
<form id="Q4Form" onSubmit="return false;">
<table class="question-choices"><tr><td><input type="radio" name="Q4option"/></td>
<td class="brown" >(a)</td><td><div class="question-choices"><!--l. 100--><math
xmlns="http://www.w3.org/1998/Math/MathML"
display="inline" ><msup><mrow
><mi
>l</mi></mrow><mrow
><msup><mrow
><mo
class="MathClass-bin">∗</mo></mrow><mrow
><mn>2</mn></mrow></msup
>
</mrow></msup
></math>
<a
id="x1-22r2"></a></div></td>
</tr><!-- part=0, cols=1, # answers = 5 -->
<tr><td><input type="radio" name="Q4option"/></td>
<td class="brown" >(b)</td><td><div class="question-choices"><!--l. 101--><math
xmlns="http://www.w3.org/1998/Math/MathML"
display="inline" ><msup><mrow
><mi
>l</mi></mrow><mrow
><msup><mrow
><mo
class="MathClass-bin">∗</mo></mrow><mrow
><mn>3</mn></mrow></msup
>
</mrow></msup
></math>
<a
id="x1-23r3"></a></div></td>
</tr><!-- part=1, cols=1, # answers = 5 -->
<tr><td><input type="radio" name="Q4option"/></td>
<td class="brown" >(c)</td><td><div class="question-choices"><!--l. 102--><math
xmlns="http://www.w3.org/1998/Math/MathML"
display="inline" ><msup><mrow
><mi
>l</mi></mrow><mrow
><msup><mrow
><mo
class="MathClass-bin">∗</mo></mrow><mrow
><mn>4</mn></mrow></msup
>
</mrow></msup
></math>
<a
id="x1-24r4"></a></div></td>
</tr><!-- part=2, cols=1, # answers = 5 -->
<tr><td><input type="radio" name="Q4option"/></td>
<td class="brown" >(d)</td><td><div class="question-choices"><!--l. 103--><math
xmlns="http://www.w3.org/1998/Math/MathML"
display="inline" ><msup><mrow
><mi
>l</mi></mrow><mrow
><msup><mrow
><mo
class="MathClass-bin">∗</mo></mrow><mrow
><mn>5</mn></mrow></msup
>
</mrow></msup
></math>
<a
id="x1-25r5"></a></div></td>
</tr><!-- part=3, cols=1, # answers = 5 -->
<tr><td><input type="radio" name="Q4option"/></td>
<td class="brown" >(e)</td><td><div class="question-choices"><!--l. 104--><math
xmlns="http://www.w3.org/1998/Math/MathML"
display="inline" ><msup><mrow
><mi
>l</mi></mrow><mrow
><msup><mrow
><mo
class="MathClass-bin">∗</mo></mrow><mrow
><mn>6</mn></mrow></msup
>
</mrow></msup
></math></div></td>
</tr><!-- part=4, cols=1, # answers = 5 -->
</table>
<p>
<input type="button" value="Check answer" name="answer" class="input-button" onClick="checkAnswer(4);"/>
<input type="button" value="Next unanswered question" class="input-button" title="Next unanswered question" name="next" onClick="nextQuestion(1);"/>
</p>
</form>
<div class="answer">
<div id="q4feedback1" class="feedback">
<em>Choice (a) is <span class="dazzle">Incorrect.</span></em>
<div></div>
</div>
<div id="q4feedback2" class="feedback">
<em>Choice (b) is <span class="dazzle">Incorrect.</span></em>
<div></div>
</div>
<div id="q4feedback3" class="feedback">
<em>Choice (c) is <span class="dazzle">Incorrect.</span></em>
<div></div>
</div>
<div id="q4feedback4" class="feedback">
<em>Choice (d) is <span class="dazzle">Incorrect.</span></em>
<div></div>
</div>
<div id="q4feedback5" class="feedback">
<em>Choice (e) is <span class="dazzle">Correct!</span></em>
<div></div>
</div></div>
</div>
<div id="question5" class="question" style="display:none;">
<span class="question-label"></span>
<div class="center"
>
Main design drivers of components</div>
Mechanical stress <!--l. 112--><math
xmlns="http://www.w3.org/1998/Math/MathML"
display="inline" ><mi
>σ</mi></math>
have a main influence on design of : <br
class="newline" />
<form id="Q5Form" onSubmit="return false;">
<table class="question-choices"><tr><td><input type="radio" name="Q5option"/></td>
<td class="brown" >(a)</td><td><div class="question-choices">Bearings <a
id="x1-28r2"></a></div></td>
</tr><!-- part=0, cols=1, # answers = 3 -->
<tr><td><input type="radio" name="Q5option"/></td>
<td class="brown" >(b)</td><td><div class="question-choices">Hydraulic jack <a
id="x1-29r3"></a></div></td>
</tr><!-- part=1, cols=1, # answers = 3 -->
<tr><td><input type="radio" name="Q5option"/></td>
<td class="brown" >(c)</td><td><div class="question-choices">Brushless motor</div></td>
</tr><!-- part=2, cols=1, # answers = 3 -->
</table>
<p>
<input type="button" value="Check answer" name="answer" class="input-button" onClick="checkAnswer(5);"/>
<input type="button" value="Next unanswered question" class="input-button" title="Next unanswered question" name="next" onClick="nextQuestion(1);"/>
</p>
</form>
<div class="answer">
<div id="q5feedback1" class="feedback">
<em>Choice (a) is <span class="dazzle">Incorrect.</span></em>
<div></div>
</div>
<div id="q5feedback2" class="feedback">
<em>Choice (b) is <span class="dazzle">Incorrect.</span></em>
<div></div>
</div>
<div id="q5feedback3" class="feedback">
<em>Choice (c) is <span class="dazzle">Correct!</span></em>
<div></div>
</div></div>
</div>
<div id="question6" class="question" style="display:none;">
<span class="question-label"></span>
<div class="center"
>
Main design drivers of components</div>
Temperature <!--l. 124--><math
xmlns="http://www.w3.org/1998/Math/MathML"
display="inline" ><mi
>θ</mi></math>
and losses have a main influence on design of : <br
class="newline" />
<form id="Q6Form" onSubmit="return false;">
<table class="question-choices"><tr><td><input type="radio" name="Q6option"/></td>
<td class="brown" >(a)</td><td><div class="question-choices">Bearings <a
id="x1-32r2"></a></div></td>
</tr><!-- part=0, cols=1, # answers = 3 -->
<tr><td><input type="radio" name="Q6option"/></td>
<td class="brown" >(b)</td><td><div class="question-choices">Hydraulic jack <a
id="x1-33r3"></a></div></td>
</tr><!-- part=1, cols=1, # answers = 3 -->
<tr><td><input type="radio" name="Q6option"/></td>
<td class="brown" >(c)</td><td><div class="question-choices">Brushless motor</div></td>
</tr><!-- part=2, cols=1, # answers = 3 -->
</table>
<p>
<input type="button" value="Check answer" name="answer" class="input-button" onClick="checkAnswer(6);"/>
<input type="button" value="Next unanswered question" class="input-button" title="Next unanswered question" name="next" onClick="nextQuestion(1);"/>
</p>
</form>
<div class="answer">
<div id="q6feedback1" class="feedback">
<em>Choice (a) is <span class="dazzle">Incorrect.</span></em>
<div></div>
</div>
<div id="q6feedback2" class="feedback">
<em>Choice (b) is <span class="dazzle">Correct!</span></em>
<div></div>
</div>
<div id="q6feedback3" class="feedback">
<em>Choice (c) is <span class="dazzle">Incorrect.</span></em>
<div></div>
</div></div>
</div>
<div id="question7" class="question" style="display:none;">
<span class="question-label"></span>
<div class="center"
>
Mechanics: stress</div>
Recall on the Buckingham theorem : the theorem states that if there is a physically
meaningful equation involving a certain number n of physical variables, then the original
equation can be rewritten in terms of a set of p = n - k dimensionless parameters
<!--l. 136--><math
xmlns="http://www.w3.org/1998/Math/MathML"
display="inline" ><msub><mrow
><mi
>π</mi></mrow><mrow
><mn>1</mn></mrow></msub
><mo
class="MathClass-punc">,</mo><msub><mrow
><mi
>π</mi></mrow><mrow
><mn>2</mn></mrow></msub
><mo
class="MathClass-punc">,</mo><mo
class="MathClass-punc">.</mo><mo
class="MathClass-punc">.</mo><mo
class="MathClass-punc">.</mo><mo
class="MathClass-punc">,</mo><msub><mrow
><mi
>π</mi></mrow><mrow
><mi
>p</mi></mrow></msub
></math>
constructed from the original variables where k is the number of independent physical
dimensions involved <br
class="newline" />
Thanks Buckingham theorem, find evolution of stress
<!--l. 139--><math
xmlns="http://www.w3.org/1998/Math/MathML"
display="inline" ><mi
>σ</mi></math> with
geometrical dimension for a rectangular sample under a load in a three-point bending
setup : <div class="minipage"> <ul class="itemize1">
<li class="itemize">F is the load (force) at the fracture point (N)
</li>
<li class="itemize">L is the length of the support span
</li>
<li class="itemize">b is width
</li>
<li class="itemize">d is thickness</li></ul></div>
<div class="center"
>
<img
src="scaling/Picture2.png" alt="PIC" width="277" height="142" /></div>
<form id="Q7Form" onSubmit="return false;">
<table class="question-choices"><tr><td><input type="radio" name="Q7option"/></td>
<td class="brown" >(a)</td><td><div class="question-choices"><!--l. 153--><math
xmlns="http://www.w3.org/1998/Math/MathML"
display="inline" ><mi
>σ</mi><mo
class="MathClass-punc">.</mo><mi
>L</mi><mo
class="MathClass-punc">.</mo><msup><mrow
><mi
>F</mi></mrow><mrow
><mo
class="MathClass-bin">−</mo><mn>1</mn></mrow></msup
> <mo
class="MathClass-rel">=</mo> <mi
>f</mi><mrow ><mo
class="MathClass-open">(</mo><mrow><mi
>b</mi><mo
class="MathClass-punc">.</mo><msup><mrow
><mi
>L</mi></mrow><mrow
><mo
class="MathClass-bin">−</mo><mn>1</mn></mrow></msup
><mo
class="MathClass-punc">,</mo><mi
>d</mi><mo
class="MathClass-punc">.</mo><msup><mrow
><mi
>L</mi></mrow><mrow
><mo
class="MathClass-bin">−</mo><mn>1</mn></mrow></msup
></mrow><mo
class="MathClass-close">)</mo></mrow></math>
<a
id="x1-36r2"></a></div></td>
</tr><!-- part=0, cols=1, # answers = 3 -->
<tr><td><input type="radio" name="Q7option"/></td>
<td class="brown" >(b)</td><td><div class="question-choices"><!--l. 154--><math
xmlns="http://www.w3.org/1998/Math/MathML"
display="inline" ><mi
>σ</mi><mo
class="MathClass-punc">.</mo><msup><mrow
><mi
>L</mi></mrow><mrow
><mn>2</mn></mrow></msup
><mo
class="MathClass-punc">.</mo><msup><mrow
><mi
>F</mi></mrow><mrow
><mo
class="MathClass-bin">−</mo><mn>1</mn></mrow></msup
> <mo
class="MathClass-rel">=</mo> <mi
>f</mi><mrow ><mo
class="MathClass-open">(</mo><mrow><mi
>b</mi><mo
class="MathClass-punc">.</mo><msup><mrow
><mi
>L</mi></mrow><mrow
><mo
class="MathClass-bin">−</mo><mn>1</mn></mrow></msup
><mo
class="MathClass-punc">,</mo><mi
>d</mi><mo
class="MathClass-punc">.</mo><msup><mrow
><mi
>L</mi></mrow><mrow
><mo
class="MathClass-bin">−</mo><mn>1</mn></mrow></msup
></mrow><mo
class="MathClass-close">)</mo></mrow></math>
<a
id="x1-37r3"></a></div></td>
</tr><!-- part=1, cols=1, # answers = 3 -->
<tr><td><input type="radio" name="Q7option"/></td>
<td class="brown" >(c)</td><td><div class="question-choices"><!--l. 155--><math
xmlns="http://www.w3.org/1998/Math/MathML"
display="inline" ><mi
>σ</mi> <mo
class="MathClass-rel">=</mo> <mi
>f</mi><mrow ><mo
class="MathClass-open">(</mo><mrow><msup><mrow
><mi