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Question

What is the minimum coefficient of friction for a solid sphere to roll without slipping on an incined plane of inclination $$\theta$$? 


A
27 tanθ
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B
13g tanθ
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C
12 tanθ
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D
25 tanθ
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Solution

The correct option is A $$\dfrac{2}{7}$$ $$\tan{\theta}$$
Here we are talking about  minimum coefficient of friction ,this suggests that
friction acting would be $$\mu mg\cos\theta$$

taking  moment about centre,
$$\mu mg\cos\theta\times r=I\times\alpha$$
$$\implies\alpha r=\dfrac{\mu mg\cos\theta r^2}{I}$$
now from force balance,
$$mg\sin\theta-\mu mg\cos\theta=ma,$$
$$\implies a=g\sin\theta-\mu g\cos\theta$$
from condition of rolling,we know that
$$a=r\alpha$$
$$\dfrac{\mu mg\cos\theta r^2}{I}=g\sin\theta-\mu mg\cos\theta,$$
simplifying this,we get
$$\mu=\dfrac{\tan\theta}{1+\dfrac{mr^2}{I}}$$

$$I=\dfrac{2}{5}mr^2$$
putting the value of moment of inertia in exprssion of coefficient of friction,we get

$$\mu=\dfrac{2}{7}\tan\theta$$




959928_1038493_ans_f7ee57b8b6b046bbb6344546caa56615.jpg

Physics

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