CAIE M2 (Mechanics 2) 2018 June

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\includegraphics[max width=\textwidth, alt={}, center]{8dda6c21-7cb5-43b6-9a34-485bdf4042c4-10_262_732_264_705} A particle \(P\) of mass 0.2 kg is attached to one end of a light inextensible string of length 0.6 m . The other end of the string is attached to a particle \(Q\) of mass 0.3 kg . The string passes through a small hole \(H\) in a smooth horizontal surface. A light elastic string of natural length 0.3 m and modulus of elasticity 15 N joins \(Q\) to a fixed point \(A\) which is 0.4 m vertically below \(H\). The particle \(P\) moves on the surface in a horizontal circle with centre \(H\) (see diagram).

1. Calculate the greatest possible speed of \(P\) for which the elastic string is not extended.
2. Find the distance \(H P\) given that the angular speed of \(P\) is \(8 \mathrm { rad } \mathrm { s } ^ { - 1 }\).

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\includegraphics[max width=\textwidth, alt={}, center]{8dda6c21-7cb5-43b6-9a34-485bdf4042c4-12_732_581_260_774} A uniform solid cone has height 1.2 m and base radius 0.5 m . A uniform object is made by drilling a cylindrical hole of radius 0.2 m through the cone along the axis of symmetry (see diagram).

1. Show that the height of the object is 0.72 m and that the volume of the cone removed by the drilling is \(0.0352 \pi \mathrm {~m} ^ { 3 }\).
   [0pt] [The volume of a cone is \(\frac { 1 } { 3 } \pi r ^ { 2 } h\).]
2. Find the distance of the centre of mass of the object from its base.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.