A massless metal plate is placed on a horizontal tabletop lubricated with oil. The sheet is a square of side length $\mathcal{l}$ = 1.0m and the oil layer has thickness h = 1.0mm. Initially one edge of the sheet coincides with one edge of the table. The sheet is pulled outwards without rotation with a constant force F = 15 N. If coefficient of viscosity of the oil is $\eta$ = 0.2 N-s/m², how long (in second) will it take to pull half of the sheet out of the table?

In this case,

$F=\eta \frac{v}{h}\mathcal{l}\left(\mathcal{l}-x\right)$            

$\therefore v=\frac{Fh}{\eta \mathcal{l}\left(\mathcal{l}-x\right)}=\frac{dx}{dt}$            

So,     $t=\frac{3\eta {\mathcal{l}}^3}{8Fh}=5sec$

<div><div><picture><source srcset="" media=" (max-width: 500px)"><img src="https://images.shiksha.com/mediadata/images/1757496690phpRazzjb.jpeg" alt="" width="247" height="181" loading="lazy"></picture></div></div><p>In this case, </p><p><span contenteditable="false"> <math> <mrow> <mi>F</mi> <mo>=</mo> <mi>η</mi> <mfrac> <mrow> <mi>v</mi> </mrow> <mrow> <mi>h</mi> </mrow> </mfrac> <mi mathvariant="script">l</mi> <mrow> <mo> (</mo> <mrow> <mi mathvariant="script">l</mi> <mo>−</mo> <mi>x</mi> </mrow> <mo>)</mo> </mrow> </mrow> </math> </span> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &lt;!- [if gte mso 9]>&lt;xml&gt; <o:OLEObject Type="Embed" ProgID="Equation.DSMT4" ShapeID="_x0000_i1025" DrawAspect="Content" ObjectID="_1819022206"> </o:OLEObject>&lt;/xml&gt;&lt;! [endif]-&gt;&nbsp;</p><p><span contenteditable="false"> <math> <mrow> <mo>∴</mo> <mi>v</mi> <mo>=</mo> <mfrac> <mrow> <mi>F</mi> <mi>h</mi> </mrow> <mrow> <mi>η</mi> <mi mathvariant="script">l</mi> <mrow> <mo> (</mo> <mrow> <mi mathvariant="script">l</mi> <mo>−</mo> <mi>x</mi> </mrow> <mo>)</mo> </mrow> </mrow> </mfrac> <mo>=</mo> <mfrac> <mrow> <mi>d</mi> <mi>x</mi> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> </math> </span> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &lt;!- [if gte mso 9]>&lt;xml&gt; <o:OLEObject Type="Embed" ProgID="Equation.DSMT4" ShapeID="_x0000_i1026" DrawAspect="Content" ObjectID="_1819022207"> </o:OLEObject>&lt;/xml&gt;&lt;! [endif]-&gt;&nbsp;</p><p>So, &nbsp;&lt;!- [if gte mso 9]>&lt;xml&gt; <o:OLEObject Type="Embed" ProgID="Equation.DSMT4" ShapeID="_x0000_i1027" DrawAspect="Content" ObjectID="_1819022208"> </o:OLEObject>&lt;/xml&gt;&lt;! [endif]-&gt;&nbsp;<span contenteditable="false"> <math> <mrow> <mi>t</mi> <mo>=</mo> <mfrac> <mrow> <mn>3</mn> <mi>η</mi> <msup> <mrow> <mi mathvariant="script">l</mi> </mrow> <mrow> <mn>3</mn> </mrow> </msup> </mrow> <mrow> <mn>8</mn> <mi>F</mi> <mi>h</mi> </mrow> </mfrac> <mo>=</mo> <mn>5</mn> <mi>s</mi> <mi>e</mi> <mi>c</mi> </mrow> </math> </span></p>

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