Consider a uniform cubical box of side a on a rough floor that is to be moved by applying minimum possible force $F$ at $a$ point $b$ above its centre of mass (see figure). If the coefficient of friction is $\mu =0.4$ , the maximum possible value of $100\times \frac{b}{a}$  for box not to topple before moving i

21.  

<p><span contenteditable="false"> <math> <mrow> <mi mathvariant="normal"> </mi> </mrow> <mn>50.00</mn> </math> </span></p><p>For no toppling</p><p><span contenteditable="false"> <math> <mi>F</mi> <mfenced separators="|"> <mrow> <mrow> <mfrac> <mrow> <mrow> <mi>a</mi> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> </mfrac> <mo>+</mo> <mi>b</mi> </mrow> </mrow> </mfenced> <mo>≤</mo> <mi>m</mi> <mi>g</mi> <mfrac> <mrow> <mrow> <mi>a</mi> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> </mfrac> </math> </span><br>&lt;!--[endif]--&gt;&lt;!--[endif]--&gt;</p><p><span contenteditable="false"> <math> <mi>μ</mi> <mfrac> <mrow> <mrow> <mi mathvariant="normal">a</mi> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> </mfrac> <mo>+</mo> <mi>μ</mi> <mi mathvariant="normal">b</mi> <mo>≤</mo> <mfrac> <mrow> <mrow> <mi mathvariant="normal">a</mi> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> </mfrac> </math> </span><br>&lt;!--[endif]--&gt;&lt;!--[endif]--&gt;</p><p><span contenteditable="false"> <math> <mn>0.2</mn> <mi>a</mi> <mo>+</mo> <mn>0.4</mn> <mi>b</mi> <mo>≤</mo> <mn>0.5</mn> <mi>a</mi> </math> </span><br>&lt;!--[endif]--&gt;&lt;!--[endif]--&gt;</p><p><span contenteditable="false"> <math> <mn>0.4</mn> <mi>b</mi> <mo>≤</mo> <mn>0.3</mn> <mi>a</mi> </math> </span><br>&lt;!--[endif]--&gt;&lt;!--[endif]--&gt;</p><p><span contenteditable="false"> <math> <mi mathvariant="normal">b</mi> <mo>≤</mo> <mfrac> <mrow> <mrow> <mn>3</mn> <mi mathvariant="normal">a</mi> </mrow> </mrow> <mrow> <mrow> <mn>4</mn> </mrow> </mrow> </mfrac> </math> </span><br>&lt;!--[endif]--&gt;&lt;!--[endif]--&gt;</p><p><span contenteditable="false"> <math> <mi mathvariant="normal">b</mi> <mo>≤</mo> <mn>0.75</mn> <mi mathvariant="normal">a</mi> <mrow> <mi mathvariant="normal"> </mi> </mrow> </math> </span> (in limiting case)</p><p>But is not possible as maximum value of &lt;!-- [if gte msEquation 12]><m:oMath><i style='mso-bidi-font-style: normal'><span style='font-family:"Cambria Math",serif'><m:r>b</m:r></span></i></m:oMath><![endif]--&gt;&lt;!-- [if !msEquation]--&gt;&lt;!-- [if gte vml 1]><v:shape id="_x0000_i1025" type="#_x0000_t75" style='width:6pt;height:13pt'> <v:imagedata src="file:///C:/Users/ALOK~1.SIN/AppData/Local/Temp/msohtmlclip1/01/clip_image015.png" o:title="" chromakey="white"/></v:shape><![endif]--&gt;&lt;!-- [if !vml]--&gt;<img src="file:///C:/Users/ALOK~1.SIN/AppData/Local/Temp/msohtmlclip1/01/clip_image016.png" width="8" height="17">&lt;!--[endif]--&gt;&lt;!--[endif]--&gt;&nbsp;can be equal to &lt;!-- [if gte msEquation 12]><m:oMath><span style='font-family:"Cambria Math",serif'><m:r><m:rPr><m:scr m:val="roman"/><m:sty m:val="p"/></m:rPr>0.5</m:r><m:r><i style='mso-bidi-font-style:normal'>a</i></m:r></span></m:oMath><![endif]--&gt;&lt;!-- [if !msEquation]--&gt;&lt;!-- [if gte vml 1]><v:shape id="_x0000_i1025" type="#_x0000_t75" style='width:21pt;height:13pt'> <v:imagedata src="file:///C:/Users/ALOK~1.SIN/AppData/Local/Temp/msohtmlclip1/01/clip_image017.png" o:title="" chromakey="white"/></v:shape><![endif]--&gt;&lt;!-- [if !vml]--&gt;<img src="file:///C:/Users/ALOK~1.SIN/AppData/Local/Temp/msohtmlclip1/01/clip_image018.png" width="28" height="17">&lt;!--[endif]--&gt;&lt;!--[endif]--&gt;&nbsp;only.</p><p><span contenteditable="false"> <math> <mo>∴</mo> <msub> <mrow> <mrow> <mfenced separators="|"> <mrow> <mrow> <mn>100</mn> <mfrac> <mrow> <mrow> <mi mathvariant="normal">b</mi> </mrow> </mrow> <mrow> <mrow> <mi mathvariant="normal">a</mi> </mrow> </mrow> </mfrac> </mrow> </mrow> </mfenced> </mrow> </mrow> <mrow> <mrow> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">x</mi> </mrow> </mrow> </msub> <mo>=</mo> <mn>50.00</mn> </math> </span><br>&lt;!--[endif]--&gt;&lt;!--[endif]--&gt;</p>

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