2. [The centre of mass of a uniform hollow cone of height \(h\) is \(\frac { 1 } { 3 } h\) above the base on the line from the centre of the base to the vertex.]
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\caption{Figure 1}
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A marker for the route of a charity walk consists of a uniform hollow cone fixed on to a uniform solid cylindrical ring, as shown in Figure 1. The hollow cone has base radius \(r\), height \(9 h\) and mass \(m\). The solid cylindrical ring has outer radius \(r\), height \(2 h\) and mass \(3 m\). The marker stands with its base on a horizontal surface.
- Find, in terms of \(h\), the distance of the centre of mass of the marker from the horizontal surface.
When the marker stands on a plane inclined at \(\arctan \frac { 1 } { 12 }\) to the horizontal it is on the point of toppling over. The coefficient of friction between the marker and the plane is large enough to be certain that the marker will not slip.
- Find \(h\) in terms of \(r\).