6.
Figure 2
\includegraphics[max width=\textwidth, alt={}, center]{c4b453e7-8a32-458b-8041-58c9e4ef9533-5_691_1067_241_584}
A uniform solid cylinder has radius \(2 a\) and height \(\frac { 3 } { 2 } a\). A hemisphere of radius \(a\) is removed from the cylinder. The plane face of the hemisphere coincides with the upper plane face of the cylinder, and the centre \(O\) of the hemisphere is also the centre of this plane face, as shown in Fig. 2. The remaining solid is \(S\).
- Find the distance of the centre of mass of \(S\) from \(O\).
(6)
The lower plane face of \(S\) rests in equilibrium on a desk lid which is inclined at an angle \(\theta\) to the horizontal. Assuming that the lid is sufficiently rough to prevent \(S\) from slipping, and that \(S\) is on the point of toppling when \(\theta = \alpha\), - find the value of \(\alpha\).
(3)
Given instead that the coefficient of friction between \(S\) and the lid is 0.8 , and that \(S\) is on the point of sliding down the lid when \(\theta = \beta\), - find the value of \(\beta\).
(3)