Question 135: Laws of Motion

Question

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 :

Accepted Answer

Let, t
1
and t
2
are time taken to move on the smooth and rough surface for smooth surface,
S = $${1 \over 2}$$ g sin45
o
$$t_1^2$$
$$ \Rightarrow $$$$\,\,\,\,$$ t
1
= $$\sqrt {{{2\sqrt 2 S} \over g}} $$
For rough surface,
S = $${1 \over 2}$$ g (sin45
o
$$-$$ $$\mu $$
k
cos45
o
) $$t_2^2$$
$$ \Rightarrow $$$$\,\,\,\,$$ t
2
= $$\sqrt {{{2\sqrt 2 S} \over {g\left( {1 - {\mu _k}} ight)}}} $$
Here $${\mu _k}$$ = Kinetic friction.
According to question,
t
2
= n t
1
$$ \Rightarrow $$$$\,\,\,\,$$ $${{2\sqrt 2 \,S} \over {g\left( {1 - {\mu _k}} ight)}}$$ = n
2
$$ 	imes $$ $${{2\sqrt 2 \,S} \over g}$$
$$ \Rightarrow $$$$\,\,\,\,$$ 1 $$-$$ $$\mu $$
k
= $${1 \over {{n^2}}}$$
$$ \Rightarrow $$$$\,\,\,\,$$ $${\mu _k}$$ = 1 $$-$$ $${1 \over {{n^2}}}$$

Answer

$${1 \over {2 - {n^2}}}$$

Answer

$$1 - {1 \over {{n^2}}}$$

Answer

$$\sqrt {1 - {1 \over {{n^2}}}} $$

Answer

$$\sqrt {{1 \over {1 - {n^2}}}} $$

Laws of Motion

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 :

Choose the correct answer: