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Force on a Wooden Cube

Let's discuss a beautiful Problem, useful for Physics Olympiad based on Force on a Wooden Cube.

The Problem: Force on a Wooden Cube

A wooden cube is placed on a rough horizontal table. A force is applied to the cube. Gradually the force is increased. Whether the cube slides before toppling or topples before sliding is independent of
(a) the position of the point of application of the force
(b) the length of the edge of the cube
(c) mass of the cube
(d) coefficient of friction between the cube and the table

Solution:

Let us consider the mass of the cube to be taken as m. For the cube to slide the force F must be greater than \(\mu mg\). The block will topple if $$ Fr>mg(a/2)$$

$$\Rightarrow F>mg(a/2r) $$( r being the distance from the ground level at which the force is applied to the cube)
Now, for the cube to topple before sliding
$$ \mu mg<mg(a/2r)$$ $$ \Rightarrow \mu g<g(a/2r)$$
For the cube to slide before toppling
$$ \mu>g(a/2r)$$
Hence, whether the cube slides before toppling or topples before sliding is independent of mass.

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