A uniform cylinder of mass M and radius R is to be pulled over a...

A uniform cylinder of mass $M$ and radius $R$ is to be pulled over a step of height a $\left(a < R\right)$ 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 uniform cylinder of mass M and radius R is to be pulled over a step of height a (a

Options

AMg\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 $Mg\sqrt{1 - \dfrac{{a}^{2}}{{R}^{2}}}$

Correct answer

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

Step-by-step solution

A uniform cylinder of mass M and radius R is to be pulled over a step of height a (a

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

$FR > Mgcos\thetaR$

$F > Mgcos\theta$

$F > Mg\dfrac{\sqrt{{R}^{2} - ( R - a{)}^{2}}}{R}$ .

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

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