M3 January 2010 Q3

EdexcelOld spec10 marksCentres of Mass

3.

Figure 1: bowl formed from a solid hemisphere of radius r and centre O with a concentric hemisphere of radius two-thirds r removed
Figure 1

A bowl \(B\) consists of a uniform solid hemisphere, of radius \(r\) and centre \(O\), from which is removed a solid hemisphere, of radius \(\dfrac{2}{3}r\) and centre \(O\), as shown in Figure 1.

(a) Show that the distance of the centre of mass of \(B\) from \(O\) is \(\dfrac{65}{152}r\). (5)
Figure 2: the bowl resting with point C of its curved surface on a horizontal plane, particle at P on the outer rim, OP at angle theta to the horizontal
Figure 2

The bowl \(B\) has mass \(M\). A particle of mass \(kM\) is attached to a point \(P\) on the outer rim of \(B\). The system is placed with a point \(C\) on its outer curved surface in contact with a horizontal plane. The system is in equilibrium with \(P\), \(O\) and \(C\) in the same vertical plane. The line \(OP\) makes an angle \(\theta\) with the horizontal as shown in Figure 2. Given that

\(\tan\theta = \dfrac{4}{5}\),

(b) find the exact value of \(k\). (5)
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