A uniform rectangular block of mass M has base dimensions L × W and height H . It is placed on a rough horizontal surface. A horizontal force F is applied perpendicular to the face of dimension W × H

Question

A uniform rectangular block of mass M has base dimensions L × W and height H . It is placed on a rough horizontal surface. A horizontal force F is applied perpendicular to the face of dimension W × H (i.e., along the length L ) at a height h from the base. If the coefficient of static friction is μ s ​ , what is the minimum height h at which the force should be applied so that the block topples before it slides?

TestHub · Accepted Answer

For the block to topple, the torque due to the applied force $F$ about the edge must exceed the restoring torque due to weight $M g$ . Toppling occurs when $F h > M g (L /2)$ . The minimum force required to topple is $F_{t o ppl e} = \frac{M gL}{2 h}$ .

For the block to slide, the applied force $F$ must exceed the maximum static friction force. Sliding occurs when $F > μ_{s} M g$ . The minimum force required to slide is $F_{s l i d e} = μ_{s} M g$ . For the block to topple before sliding, we must have $F_{t o ppl e} < F_{s l i d e}$ . $\frac{M gL}{2 h} < μ_{s} M g ⟹ \frac{L}{2 h} < μ_{s} ⟹ h > \frac{L}{2 μ _{s}}$ . Thus, the minimum height $h$ for the block to topple before sliding is $\frac{L}{2 μ _{s}}$ .

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