Question 303 of 308

A uniform cylinder of mass $M$ and radius $R$ is to be pulled over a step of height a $\left(a\lt R\right)$ by applying a force $F$ at its centre $'O'$ perpendicular to the plane through the axes of the cylinder on the edge of the step (see figure). The minimum value of $F$ required is :

A$Mg\sqrt{1-{\left(\dfrac{R-a}{R}\right)}^{2}}$B$Mg\sqrt{{\left(\dfrac{R}{R-a}\right)}^{2}-1}$C$Mg\dfrac{a}{R}$D$Mg\sqrt{1-\dfrac{{a}^{2}}{{R}^{2}}}$

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Answer: (A) $Mg\sqrt{1-{\left(\dfrac{R-a}{R}\right)}^{2}}$

$Mg\sqrt{1-{\left(\dfrac{R-a}{R}\right)}^{2}}$
$FR\gt Mgcos\theta R$
$F\gt Mgcos\theta$
$F\gt Mg\dfrac{\sqrt{{R}^{2}-(R-a{)}^{2}}}{R}$.
$\Rightarrow Mg\sqrt{1-{\left(\dfrac{R-a}{R}\right)}^{2}}$