AS October 2020 Q1

EdexcelCurrent spec15 marksCentres of Mass

1.

Figure 1: rectangle ABCD with A bottom left, B bottom right, C top right and D top left; AB = 2a and AD = a
Figure 1

Figure 1 shows a uniform rectangular lamina \(ABCD\) with \(AB = 2a\) and \(AD = a\)
The mass of the lamina is \(6m\).

A particle of mass \(2m\) is attached to the lamina at \(A\), a particle of mass \(m\) is attached to the lamina at \(B\) and a particle of mass \(3m\) is attached to the lamina at \(D\), to form a loaded lamina \(L\) of total mass \(12m\).

(a) Write down the distance of the centre of mass of \(L\) from \(AB\). You must give a reason for your answer. (2)
(b) Show that the distance of the centre of mass of \(L\) from \(AD\) is \(\dfrac{2a}{3}\) (3)

A particle of mass \(km\) is now also attached to \(L\) at \(D\) to form a new loaded lamina \(N\).

(c) Show that the distance of the centre of mass of \(N\) from \(AB\) is \(\dfrac{(k+6)a}{(k+12)}\) (4)

When \(N\) is freely suspended from \(A\) and is hanging in equilibrium, the side \(AB\) makes an angle \(\alpha\) with the vertical, where \(\tan\alpha = \dfrac{3}{2}\)

(d) Find the value of \(k\). (6)
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