JEE MAIN - Physics (2018 - 15th April Morning Slot - No. 20)
A given object takes n times more time to slide down a $${45^ \circ }$$ rough inclined plane as it takes to slide down a perfectly smooth $${45^ \circ }$$ incline. The coefficient of kinetic friction between the object and the incline is :
$${1 \over {2 - {n^2}}}$$
$$1 - {1 \over {{n^2}}}$$
$$\sqrt {1 - {1 \over {{n^2}}}} $$
$$\sqrt {{1 \over {1 - {n^2}}}} $$
Explanation
Let, t1 and t2 are time taken to move on the smooth and rough surface for smooth surface,
S = $${1 \over 2}$$ g sin45o $$t_1^2$$
$$ \Rightarrow $$$$\,\,\,\,$$ t1 = $$\sqrt {{{2\sqrt 2 S} \over g}} $$
For rough surface,
S = $${1 \over 2}$$ g (sin45o $$-$$ $$\mu $$k cos45o) $$t_2^2$$
$$ \Rightarrow $$$$\,\,\,\,$$ t2 = $$\sqrt {{{2\sqrt 2 S} \over {g\left( {1 - {\mu _k}} \right)}}} $$
Here $${\mu _k}$$ = Kinetic friction.
According to question,
t2 = n t1
$$ \Rightarrow $$$$\,\,\,\,$$ $${{2\sqrt 2 \,S} \over {g\left( {1 - {\mu _k}} \right)}}$$ = n2 $$ \times $$ $${{2\sqrt 2 \,S} \over g}$$
$$ \Rightarrow $$$$\,\,\,\,$$ 1 $$-$$ $$\mu $$k = $${1 \over {{n^2}}}$$
$$ \Rightarrow $$$$\,\,\,\,$$ $${\mu _k}$$ = 1 $$-$$ $${1 \over {{n^2}}}$$
S = $${1 \over 2}$$ g sin45o $$t_1^2$$
$$ \Rightarrow $$$$\,\,\,\,$$ t1 = $$\sqrt {{{2\sqrt 2 S} \over g}} $$
For rough surface,
S = $${1 \over 2}$$ g (sin45o $$-$$ $$\mu $$k cos45o) $$t_2^2$$
$$ \Rightarrow $$$$\,\,\,\,$$ t2 = $$\sqrt {{{2\sqrt 2 S} \over {g\left( {1 - {\mu _k}} \right)}}} $$
Here $${\mu _k}$$ = Kinetic friction.
According to question,
t2 = n t1
$$ \Rightarrow $$$$\,\,\,\,$$ $${{2\sqrt 2 \,S} \over {g\left( {1 - {\mu _k}} \right)}}$$ = n2 $$ \times $$ $${{2\sqrt 2 \,S} \over g}$$
$$ \Rightarrow $$$$\,\,\,\,$$ 1 $$-$$ $$\mu $$k = $${1 \over {{n^2}}}$$
$$ \Rightarrow $$$$\,\,\,\,$$ $${\mu _k}$$ = 1 $$-$$ $${1 \over {{n^2}}}$$
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