8 [In this question, you may use the fact that the volume of a right circular cone of base radius \(r\) and height \(h\) is \(\frac { 1 } { 3 } \pi r ^ { 2 } h\).]
- By using integration, show that the centre of mass of a uniform solid right circular cone of height \(h\) and base radius \(r\) is at a distance \(\frac { 3 } { 4 } h\) from the vertex.
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\caption{Fig. 8}
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Fig. 8 shows the side view of a toy formed by joining a uniform solid circular cylinder of radius \(r\) and height \(2 r\) to a uniform solid right circular cone, made of the same material as the cylinder, of radius \(r\) and height \(r\).
The toy is placed on a horizontal floor with the curved surface of the cone in contact with the floor. - Determine whether the toy will topple.
- Explain why it is not necessary to know whether the floor is rough or smooth in answering part (b).
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\includegraphics[alt={},max width=\textwidth]{cce64530-6284-409d-867a-e26c27d3e50a-06_397_1036_264_255}
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\caption{Fig. 9}
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Fig. 9 shows a uniform rod AB of length \(2 a\) and weight \(8 W\) which is smoothly hinged at the end A to a point on a fixed horizontal rough bar. A small ring of weight \(W\) is threaded on the bar and is connected to the rod at B by a light inextensible string of length \(2 a\). The system is in equilibrium with the rod inclined at an angle \(\theta\) to the horizontal.