M3 June 2011 Q3

EdexcelOld spec9 marksCentres of Mass

3.

Figure 2: cylinder of height 5l and radius 3l with a hemisphere of radius 3l on top, O the centre of the common face
Figure 2

A solid consists of a uniform solid right cylinder of height \(5l\) and radius \(3l\) joined to a uniform solid hemisphere of radius \(3l\). The plane face of the hemisphere coincides with a circular end of the cylinder and has centre \(O\), as shown in Figure 2.

The density of the hemisphere is twice the density of the cylinder.

(a) Find the distance of the centre of mass of the solid from \(O\). (5)
Figure 3: the solid standing on its circular face on a plane inclined at theta degrees
Figure 3

The solid is now placed with its circular face on a plane inclined at an angle \(\theta^\circ\) to the horizontal, as shown in Figure 3. The plane is sufficiently rough to prevent the solid slipping. The solid is on the point of toppling.

(b) Find the value of \(\theta\). (4)
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