Question 4

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:

Torque equilibrium about the step corner: $$\tau_F = \tau_{Mg}$$

Perpendicular distance for $$F$$: $$r_{\perp, F} = R$$

Perpendicular distance for $$Mg$$: $$r_{\perp, Mg} = \sqrt{R^2 - (R-a)^2} = R\sqrt{1 - \left(\frac{R-a}{R}\right)^2}$$

Equating torques: $$F \cdot R = Mg \cdot R\sqrt{1 - \left(\frac{R-a}{R}\right)^2}$$

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

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