A smooth block is released at rest on a 45° incline and then slides a distance d. The time taken to slide is n times as much to slide on rough incline than on a smooth incline. The coefficient of friction is

$straight mu subscript straight k space equals space 1 minus 1 over straight n squared$
$straight mu subscript straight k space equals space square root of 1 minus 1 over straight n squared end root$
$straight mu subscript straight s space equals space 1 minus 1 over straight n squared$
$straight mu subscript s space equals space square root of 1 minus 1 over straight n squared end root$

Solution

straight mu subscript straight k space equals space 1 minus 1 over straight n squared

straight d space equals space 1 half fraction numerator straight g over denominator square root of 2 end fraction straight t subscript 1 superscript 2 straight d equals space 1 half fraction numerator straight g over denominator square root of 2 end fraction left parenthesis 1 minus straight mu subscript straight k right parenthesis straight t subscript 2 superscript 2 fraction numerator straight t subscript 2 superscript 2 over denominator straight t subscript 1 superscript 2 end fraction space equals space straight n squared space equals fraction numerator 1 over denominator 1 minus straight mu subscript straight k end fraction

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