Laws Of Motion Question 337
Question: A given object takes n times as much time to slide down a $ 45{}^\circ $ rough incline as it takes to slide down a perfectly smooth$ 45{}^\circ $ incline. The coefficient of friction between the object and the incline is
Options:
A) $ ( 1-1/n^{2} ) $
B) $ 1/( 1-n^{2} ) $
C) $ \sqrt{( 1-1/n^{2} )} $
D) $ 1/\sqrt{( 1-n^{2} )} $
Show Answer
Answer:
Correct Answer: A
Solution:
[a] $ Smooth\text{ }surface,,,S=\frac{1}{2}g,,\sin ,,\theta ,,{{t} _{1}}^{2}….(i)$
$ For \text{ rough }surface,\text a=g{}sin,\theta -\mu ,,\cos \theta \text{ }{{\text{t}} _{\text{2}}}^{\text{2}}$
$\therefore ,,s=\frac{1}{2}g\left( sin\theta -\mu cos\theta \right){{t} _{2}}^{2}$ …(ii)
From (i) and (ii), $\therefore ,,s=\frac{1}{2}gsin\theta {{t} _{1}}^{2}=\frac{1}{2}\left( sin\theta -\mu cos\theta \right){{t} _{2}}^{2}$ Given $\theta ={{45}^{o}}\therefore {{t} _{1}}^{2}=\left( 1-\mu \right){{t} _{2}}^{2}$
Also, given that. ${{t} _{2}}=n{{t} _{1}}\therefore {{t} _{1}}^{2}=(1-\mu ){{n}^{2}}{{t} _{1}}^{2}$
$\frac{1}{{{n}^{2}}}=1-\mu \therefore \mu =\left( 1-\frac{1}{{{n}^{2}}} \right)$
