Sol. To overturn a cube about one edge (e.g. AB) or a cylinder, either must be turned so that their diagonal planes ABCD or KLMN (Fig. a and b) occupy a vertical position. For this work must be done to raise the centre of gravity of the body and the work will be greater the higher the centre of gravity has to be raised (since the weight of cube and cylinder are the same)

If the diagonal plane of the cube or cylinder is to occupy a vertical position, the diagonal AD must be rotated through an angle α about the edge AB and the diagonal plane of the cylinder must be turned through an angle β ; the cube's centre of gravity will then rise

Δ h 1 = ,

and the cylinder's centre of gravity will rise

Δ h 2 =

(Fig. c and d) . Since the heights and weights of the cube and cylinder are equal and their material is the same, their base areas are also equal, i.e.,

h 2 = π r 2 ,

where r is the radius of the base of the cylinder. Plainly for the cube α = 45º, For the cylinder

2r = h tan β

or 4r 2 = h 2 tan 2 β .

Substituting for h 2 , we get

4r 2 = π r 2 tan 2 β

or tan 2 β = > 1,

i.e., β > 45º. Therefore cos β < cos α , and therefore Δ h 2 > Δ h 1 and it is harder to overturn the cylinder than the cube about one edge.

If the attempt was made to overturn the cube about a corner (instead of about the edge), then it would have to be turned so that the diagonal AC took up a vertical position. For this the cube must be turned through an angle γ , formed by this diagonal and the cube's height (Fig. e) Then tan 2 γ = 2. Since tan 2 β = 4/ π < 2, γ > β . Therefore cos β > cos γ and it is harder to overturn a cube about one of its corners than to overturn a cylinder.