When a body slides down from rest along a smooth inclined plane making an angle of 30° with the horizontal, it takes time T. When the same body slides down from the rest along a rough inclined plane making the same angle and though the same distance, it takes time aT, where a is a constant greater than 1. The co-efficient of friction between the body and the rough plane is $\frac{1}{\sqrt[]{x}} (\frac{α^{2} - 1}{α})$ where x =___________.

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alok kumar singh | Contributor-Level 10

a month ago

for smooth surface

$a = g s i n 30 ° = \frac{g}{2}$

$S_{1} = u t + \frac{1}{2} a t^{2}$

$S_{1} = \frac{1}{2} \frac{g}{2} t^{2} = \frac{g}{4} t^{2} . . . . . . . (i)$

for rough Surface

$S = \frac{1}{2} \frac{g}{2} (1 - μ \sqrt[]{3}) α^{2} t^{2} . . . . . . . . (i i)$

By (i) and (ii)

$μ = \frac{1}{\sqrt[]{3}} (\frac{α^{2} - 1}{α^{2}}) x = \sqrt[]{3}$

for smooth surface <math> <mrow> <mi>a</mi> <mo>=</mo> <mi>g</mi> <mi>s</mi> <mi>i</mi> <mi>n</mi> <mn>3</mn> <mn>0</mn> <mo>°</mo> <mo>=</mo> <mfrac> <mrow> <mi>g</mi> </mrow> <mrow> <mn>2</mn> </mrow> </mfrac> </mrow> </math> <math> <mrow> <msub> <mrow> <mi>S</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mi>u</mi> <mi>t</mi> <mo>+</mo> <mfrac> <mrow> <mn>1</mn> </mrow> <mrow> <mn>2</mn> </mrow> </mfrac> <mi>a</mi> <msup> <mrow> <mi>t</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> </mrow> </math>          <math> <mrow> <msub> <mrow> <mi>S</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mfrac> <mrow> <mn>1</mn> </mrow> <mrow> <mn>2</mn> </mrow> </mfrac> <mfrac> <mrow> <mi>g</mi> </mrow> <mrow> <mn>2</mn> </mrow> </mfrac> <msup> <mrow> <mi>t</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mfrac> <mrow> <mi>g</mi> </mrow> <mrow> <mn>4</mn> </mrow> </mfrac> <msup> <mrow> <mi>t</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mrow> <mo>(</mo> <mrow> <mi>i</mi> </mrow> <mo>)</mo> </mrow> </mrow> </math>               for rough Surface <math> <mrow> <mi>S</mi> <mo>=</mo> <mfrac> <mrow> <mn>1</mn> </mrow> <mrow> <mn>2</mn> </mrow> </mfrac> <mfrac> <mrow> <mi>g</mi> </mrow> <mrow> <mn>2</mn> </mrow> </mfrac> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mi>μ</mi> <mroot> <mrow> <mn>3</mn> </mrow> <mrow></mrow> </mroot> </mrow> <mo>)</mo> </mrow> <msup> <mrow> <mi>α</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <msup> <mrow> <mi>t</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mrow> <mo>(</mo> <mrow> <mi>i</mi> <mi>i</mi> </mrow> <mo>)</mo> </mrow> </mrow> </math>               By (i) and (ii) <math> <mrow> <mi>μ</mi> <mo>=</mo> <mfrac> <mrow> <mn>1</mn> </mrow> <mrow> <mroot> <mrow> <mn>3</mn> </mrow> <mrow></mrow> </mroot> </mrow> </mfrac> <mrow> <mo>(</mo> <mrow> <mfrac> <mrow> <msup> <mrow> <mi>α</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> <mrow> <msup> <mrow> <mi>α</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mi>x</mi> <mo>=</mo> <mroot> <mrow> <mn>3</mn> </mrow> <mrow></mrow> </mroot> </mrow> </math>            <div><div><picture><source srcset="" media="(max-width: 500px)"><img src="https://images.shiksha.com/mediadata/images/1754477303phpJvjzrG.jpeg" alt="" width="297" height="198" loading="lazy"></picture></div></div>

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