First slide

Rolling motion

Question

What should be the minimum coefficient of static friction between the plane and the cylinder for the cylinder not to slip on an inclined plane?

Moderate

A

$\frac{1}{3} \sin θ$
B

$\frac{1}{3} \tan θ$
C

$\frac{2}{3} \sin θ$
D

$\frac{2}{3} \tan θ$

Solution

Equation of motion
$Mg \sin θ - f = Ma - - - - - - (i)$
Also fR = $τ = lα = {Mk}^{2} \frac{a}{R} - - - - - (ii)$
But $a = \frac{g sinθ}{1 + \frac{k^{2}}{R^{2}}} - - - - - - - - - - (iii)$
Putting value of a in equation (ii)
$f = (\frac{{Mk}^{2}}{R^{2}}) \frac{g \sin θ}{1 + \frac{k^{2}}{R^{2}}}$
For cylinder: ${Mk}^{2} = I = \frac{1}{2} {MR}^{2}$
$k^{2} = \frac{1}{2} R^{2}; f = \frac{M (\frac{1}{2}) g \sin θ}{(1 + \frac{1}{2})} = \frac{1}{3} Mg \sin θ$
In case of static friction, $f_{s} = μN = μMg \cos θ$
$\frac{1}{3} Mg \sin θ = μMg \cos θ \Rightarrow μ = \frac{1}{3} tanθ$

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