M3 January 2007 Q6

EdexcelOld spec13 marksCentres of Mass

6.

Figure 4: shaded region R under the curve between x = 1 and x = 2
Figure 4

The shaded region \(R\) is bounded by the curve with equation \(y = \dfrac{1}{2x^2}\), the \(x\)-axis and the lines \(x = 1\) and \(x = 2\), as shown in Figure 4. The unit of length on each axis is 1 m. A uniform solid \(S\) has the shape made by rotating \(R\) through 360\(^\circ\) about the \(x\)-axis.

(a) Show that the centre of mass of \(S\) is \(\tfrac{2}{7}\) m from its larger plane face. (6)
Figure 5: trophy T made of hemisphere H of radius one half m on top of the solid S
Figure 5

A sporting trophy \(T\) is a uniform solid hemisphere \(H\) joined to the solid \(S\). The hemisphere has radius \(\tfrac{1}{2}\) m and its plane face coincides with the larger plane face of \(S\), as shown in Figure 5. Both \(H\) and \(S\) are made of the same material.

(b) Find the distance of the centre of mass of \(T\) from its plane face. (7)
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