M3 June 2010 Q4

EdexcelOld spec10 marksCentres of Mass

4.

Figure 3: cylinder of height 6l with a cone of height 4l removed from one end, O at the centre of the open face
Figure 3

A container is formed by removing a right circular solid cone of height \(4l\) from a uniform solid right circular cylinder of height \(6l\). The centre \(O\) of the plane face of the cone coincides with the centre of a plane face of the cylinder and the axis of the cone coincides with the axis of the cylinder, as shown in Figure 3. The cylinder has radius \(2l\) and the base of the cone has radius \(l\).

(a) Find the distance of the centre of mass of the container from \(O\). (6)
Figure 4: the container standing on a plane inclined at theta degrees with its open face uppermost
Figure 4

The container is placed on a plane which is inclined at an angle \(\theta^\circ\) to the horizontal. The open face is uppermost, as shown in Figure 4. The plane is sufficiently rough to prevent the container from sliding. The container is on the point of toppling.

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