  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  B
13g tanθ  C
12 tanθ  D
25 tanθ  Solution

The correct option is A $$\dfrac{2}{7}$$ $$\tan{\theta}$$Here we are talking about  minimum coefficient of friction ,this suggests thatfriction 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$$\mu=\dfrac{2}{7}\tan\theta$$ Physics

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