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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 θ?

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 27 tanθHere we are talking about minimum coefficient of friction ,this suggests thatfriction acting would be μmgcosθtaking moment about centre,μmgcosθ×r=I×α⟹αr=μmgcosθr2Inow from force balance,mgsinθ−μmgcosθ=ma,⟹a=gsinθ−μgcosθfrom condition of rolling,we know thata=rαμmgcosθr2I=gsinθ−μmgcosθ,simplifying this,we getμ=tanθ1+mr2II=25mr2putting the value of moment of inertia in exprssion of coefficient of friction,we getμ=27tanθ

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