A uniform solid sphere of radius R and mass m rolls down an inclined plane. The coefficient of friction between the sphere and the inclined plane is μ then maximum value of θ for pure rolling is
A
tan−1(3μ2)
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B
tan−1(7μ2)
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C
tan−1(5μ3)
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D
tan−1(7μ3)
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Solution
The correct option is Btan−1(7μ2) F.B.D of the uniform solid sphere
N=mgcosθ...........(1) mg sin θ−f=ma.........(2) fR=Iα=25mR2×aR
[ for uniform rolling, a=αr ] ∴f=25ma.........(3)
On putting (3) in (2) mgsinθ=75ma a=57g sin θ..........(4)
On putting (4) in (3) f=27mg sin θ
Now, for critical case, f≤fsmax=μN ⇒27mg sin θ≤μmg cos θ
[ from (1) ] ⇒θ≤tan −1(7μ2)