A2 October 2020 Q2

EdexcelCurrent spec10 marksCentres of Mass

2.

Figure 1: the curve y = 8e^(-x) with the region R shaded between the curve and the x-axis from x = ln 2 to x = ln 5
Figure 1
Figure 2: lamina ABCD the same shape as R: AB along the bottom, AD and BC vertical with AD taller than BC, and the curved edge DC
Figure 2

A uniform plane figure \(R\), shown shaded in Figure 1, is bounded by the \(x\)-axis, the line with equation \(x = \ln 5\), the curve with equation \(y = 8\mathrm{e}^{-x}\) and the line with equation \(x = \ln 2\). The unit of length on each axis is one metre.

The area of \(R\) is \(2.4\ \text{m}^2\)

The centre of mass of \(R\) is at the point with coordinates \((\bar{x}, \bar{y})\).

(a) Use algebraic integration to show that \(\bar{y} = 1.4\) (4)

Figure 2 shows a uniform lamina \(ABCD\), which is the same size and shape as \(R\). The lamina is freely suspended from \(C\) and hangs in equilibrium with \(CB\) at an angle \(\theta^\circ\) to the downward vertical.

(b) Find the value of \(\theta\) (6)
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