Question

A uniform cylinder of mass $M$ and radius $R$ is to be pulled over a step of height $a(a<R)$ 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. $Mg\sqrt{1-{\left(\frac{R-a}{R}\right)}^2}$
• B. $Mg\sqrt{{\left(\frac{R}{R-a}\right)}^2-1}$
• C. $Mg\frac{a}{R}$
• D. $Mg\sqrt{1-\frac{a^2}{R^2}}$

Answer 1

The correct option is A $Mg\sqrt{1-{\left(\frac{R-a}{R}\right)}^2}$


With respect to the point of contact on the step,

For step up,

${\tau }_F\ge {\tau }_{mg}$

$\Rightarrow F\times R\ge Mg\times x$

From the figure, $x=\sqrt{R^2-(R-a{)}^2}$

$\Rightarrow F_{min}=\frac{Mg}{R}\times \sqrt{R^2-(R-a{)}^2}$

$\Rightarrow F_{min}=Mg\sqrt{1-{\left(\frac{R-a}{R}\right)}^2}$

Hence, option $\left(A\right)$ is correct.