A2 October 2021 Q3

EdexcelCurrent spec6 marksCentres of Mass

3.

Figure 2: bowl: a solid hemisphere of radius 2a with a hemisphere of radius a removed from it, both with centre O on the plane face
Figure 2

A uniform solid hemisphere \(H\) has radius \(2a\). A solid hemisphere of radius \(a\) is removed from the hemisphere \(H\) to form a bowl. The plane faces of the hemispheres coincide and the centres of the two hemispheres coincide at the point \(O\), as shown in Figure 2.

The centre of mass of the bowl is at the point \(G\).

(a) Show that \(OG = \dfrac{45a}{56}\) (4)

Figure 3 below shows a cross-section of the bowl which is resting in equilibrium with a point \(P\) on its curved surface in contact with a rough plane. The plane is inclined to the horizontal at an angle \(\alpha\) and is sufficiently rough to prevent the bowl from slipping. The line \(OG\) is horizontal and the points \(O\), \(G\) and \(P\) lie in a vertical plane which passes through a line of greatest slope of the inclined plane.

Figure 3: cross-section of the bowl with its plane face vertical and OG horizontal, resting with its curved surface touching a plane inclined at angle alpha to the horizontal at the point P
Figure 3
(b) Find the size of \(\alpha\), giving your answer in degrees to 3 significant figures. (2)
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