M3 January 2011 Q2

Figure 1: toy made of a cone of height 2r with vertex A joined to a hemisphere of radius r, O the centre of the common plane face and B on its rim — Figure 1

A toy is formed by joining a uniform solid hemisphere, of radius \(r\) and mass \(4m\), to a uniform right circular solid cone of mass \(km\). The cone has vertex \(A\), base radius \(r\) and height \(2r\). The plane face of the cone coincides with the plane face of the hemisphere. The centre of the plane face of the hemisphere is \(O\) and \(OB\) is a radius of its plane face as shown in Figure 1. The centre of mass of the toy is at \(O\).

(a) Find the value of \(k\). (4)

A metal stud of mass \(\lambda m\) is attached to the toy at \(A\). The toy is now suspended by a light string attached to \(B\) and hangs freely at rest. The angle between \(OB\) and the vertical is 30\(^\circ\).

(b) Find the value of \(\lambda\). (4)

Mark scheme (a)

Scheme	Marks
\(\begin{array}{lcc\|c}\text{Mass ratio} & 4m & km & (4 + k)m\\ \text{Dist from } O & \dfrac{3}{8}r & -\dfrac{1}{2}r & 0\end{array}\)	B1 B1
Moments about \(O\): \(\dfrac{3}{8}r \times 4m = \dfrac{1}{2}r \times km\)	M1
\(k = 3\)	A1
	(4)

Diagram: toy hanging from B with OB at 30 degrees to the vertical, weights 7mg at O and lambda mg at A, G on OA vertically below B — Mark scheme (b)

Scheme	Marks
\(\tan 30 = \dfrac{OG}{r}\)	B1
\(OG = \dfrac{\lambda}{(7 + \lambda)} \times 2r\)	M1
\(\dfrac{1}{\sqrt{3}} = \dfrac{\lambda}{(7 + \lambda)} \times 2r \times \dfrac{1}{r}\)	A1
\(7 + \lambda = 2\sqrt{3}\lambda\) \(\lambda = \dfrac{7}{\left(2\sqrt{3} - 1\right)}\) (o.e.) or 2.84	A1
	(4)
	(8 marks)