When a body slides down from rest along a smooth inclined plane making an angle of 45° with the horizontal, it takes time T. When the same body slides down from rest along a rough inclined plane making the same angle and through the same distance, it is seen to take time pT, where p is some number greater than 1. Calculate the co-efficient of friction between the body and the rough plane.

This is a Long Answer type Questions as classified in NCERT Exemplar

Explanation- as angle is 45

On smooth inclined plane acceleration will be a = gsin $\theta$

So acceleration will become a= g/ $\sqrt[]{2}$

Using equation of motion s =ut +1/2at²

S= $\frac{1}{2}\frac{g}{\sqrt[]{2}}{T}^2$

On rough inclined plane a = g (sin $\theta -\mu cos\theta$ )

= g (sin $45-\mu cos45$ )= $\frac{g(1-\mu )}{\sqrt[]{2}}$

So s=ut +1/2at²

S= 0+ $\frac{1}{2}\frac{g\left(1-\mu \right)}{\sqrt[]{2}}\left(pT\right)$ ²

Comparing two above distance

$\frac{\mathrm{}}{}$$\frac{1}{2}\frac{g\left(1-\mu \right)}{\text{exception 25:}}\left(pT\right)$²= $\frac{1}{2}\frac{g}{\text{exception 25:}}{T}^2$

So after solving we get $$$\mu =(1-1/p^2)$

<p>This is a Long Answer type Questions as classified in NCERT Exemplar</p><p><strong>Explanation-&nbsp;</strong>as angle is 45</p><p>On smooth inclined plane acceleration will be a = gsin<span contenteditable="false"> <math> <mi>θ</mi> </math> </span></p><div><div><picture><source srcset="https://images.shiksha.com/mediadata/images/articles/1745403011phpPbPVAr_480x360.jpeg" media=" (max-width: 500px)"><img src="https://images.shiksha.com/mediadata/images/articles/1745403011phpPbPVAr.jpeg" alt="" width="117" height="98"></picture></div></div><p>So acceleration will become a= g/<span contenteditable="false"> <math> <mroot> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> <mrow></mrow> </mroot> </math> </span></p><p>Using equation of motion s =ut +1/2at²</p><p>S=<span contenteditable="false"> <math> <mfrac> <mrow> <mrow> <mn>1</mn> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> </mfrac> <mfrac> <mrow> <mrow> <mi>g</mi> </mrow> </mrow> <mrow> <mrow> <mroot> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> <mrow></mrow> </mroot> </mrow> </mrow> </mfrac> <msup> <mrow> <mrow> <mi>T</mi> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> </msup> </math> </span></p><p>On rough inclined plane a = g (sin<span contenteditable="false"> <math> <mi>θ</mi> <mo>-</mo> <mi>μ</mi> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mi>θ</mi> </math> </span>)</p><p>= g (sin<span contenteditable="false"> <math> <mn>45</mn> <mo>-</mo> <mi>μ</mi> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mn>45</mn> </math> </span>)=<span contenteditable="false"> <math> <mfrac> <mrow> <mrow> <mi>g</mi> <mo> (</mo> <mn>1</mn> <mo>-</mo> <mi>μ</mi> <mo>)</mo> </mrow> </mrow> <mrow> <mrow> <mroot> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> <mrow></mrow> </mroot> </mrow> </mrow> </mfrac> </math> </span></p><p>So s=ut +1/2at²</p><p>S= 0+<span contenteditable="false"> <math> <mfrac> <mrow> <mrow> <mn>1</mn> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> </mfrac> <mfrac> <mrow> <mrow> <mi>g</mi> <mfenced separators="|"> <mrow> <mrow> <mn>1</mn> <mo>-</mo> <mi>μ</mi> </mrow> </mrow> </mfenced> </mrow> </mrow> <mrow> <mrow> <mroot> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> <mrow></mrow> </mroot> </mrow> </mrow> </mfrac> <mo> (</mo> <mi>p</mi> <mi>T</mi> <mo>)</mo> </math> </span>²</p><p>Comparing two above distance</p><p><math><mfrac><mrow><mrow><mn></mn></mrow></mrow></mfrac></math><strong><math><mfrac><mrow><mrow><mn>1</mn></mrow></mrow><mrow><mrow><mn>2</mn></mrow></mrow></mfrac><mfrac><mrow><mrow><mi>g</mi><mfenced separators="|"><mrow><mrow><mn>1</mn><mo>-</mo><mi>μ</mi></mrow></mrow></mfenced></mrow></mrow><mrow><mrow><mroot><mrow><mrow><mn>2</mn></mrow></mrow></mroot></mrow></mrow></mfrac><mo> (</mo><mi>p</mi><mi>T</mi><mo>)</mo></math>²= <math><mfrac><mrow><mrow><mn>1</mn></mrow></mrow><mrow><mrow><mn>2</mn></mrow></mrow></mfrac><mfrac><mrow><mrow><mi>g</mi></mrow></mrow><mrow><mrow><mroot><mrow><mrow><mn>2</mn></mrow></mrow></mroot></mrow></mrow></mfrac><msup><mrow><mrow><mi>T</mi></mrow></mrow><mrow><mrow><mn>2</mn></mrow></mrow></msup></math></strong></p><p>So after solving we get&nbsp;<math><mi></mi></math><strong><math><mi>μ</mi><mo>=</mo><mo> (</mo><mn>1</mn><mo>-</mo><mn>1</mn><mo>/</mo><msup><mrow><mrow><mi>p</mi></mrow></mrow><mrow><mrow><mn>2</mn></mrow></mrow></msup><mo>)</mo></math></strong></p>

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