Question 146: Laws of Motion

Question

A smooth block is released at rest on a $${45^ \circ }$$ 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

Accepted Answer

For smooth surface,
$$d = {1 \over 2}\left( {g\,\sin \,	heta } ight)t_1^2,$$
$${t_1} = \sqrt {{{2d} \over {g\,\sin \,	heta }}} ,$$
When surface is rough
$$d = {1 \over 2}\left( {g\,\sin \,	heta  - \mu g\,\cos 	heta } ight)t_2^2$$
$${t_2} = \sqrt {{{2d} \over {g\,\sin \,	heta  - \mu g\,\cos 	heta }}} $$
According to question, $${t_2} = n{t_1}$$
$$n\sqrt {{{2d} \over {g\,\sin \,	heta }}}  = \sqrt {{{2d} \over {g\,\sin \,	heta  - \mu g\,\cos 	heta }}} $$
$$n = {1 \over {\sqrt {1 - {\mu _k}} }}$$  ( as $$\cos \,{45^ \circ } = \sin \,{45^ \circ } = {1 \over {\sqrt 2 }}$$ )
$${n^2} = {1 \over {1 - {\mu _k}}}$$
or $$1 - {\mu _k} = {1 \over {{n^2}}}$$
or $${\mu _k} = 1 - {1 \over {{n^2}}}$$

Answer

$${\mu _k} = \sqrt {1 - {1 \over {{n^2}}}} $$

Answer

$${\mu _k} = 1 - {1 \over {{n^2}}}$$

Answer

$${\mu _k} = \sqrt {1 - {1 \over {{n^2}}}} $$

Answer

$${\mu _s} = 1 - {1 \over {{n^2}}}$$

Laws of Motion

A smooth block is released at rest on a ${45^ \circ } incline and then slides a distance 'd'. The time taken to slide is 'n'$ times as much to slide on rough incline than on a smooth incline. The coefficient of friction is

Choose the correct answer: