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 =___________.

3 Views | Posted a month ago

Asked by Shiksha User

1 Answer

Answered by

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>

0 Upvotes 0 Downvotes

Get authentic answers from experts, students and alumni that you won't find anywhere else

On Shiksha, get access to

65k Colleges
1.2k Exams
688k Reviews
1800k Answers

Community GuidelinesUser Points System

QuestionsDiscussionsTags

1 Answer

Learn more about...

Most viewed information

Share Your College Life Experience

Didn't find the answer you were looking for?

Need guidance on career and education? Ask our experts

Your Question