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\includegraphics[max width=\textwidth, alt={}, center]{b5b63135-6d02-4c4a-835a-9834d2852d6b-4_739_860_1114_616} A conical shell has radius 6 m and height 8 m . The shell, with its vertex \(V\) downwards, is rotating about its vertical axis. A particle, of mass 0.4 kg , is in contact with the rough inner surface of the shell. The particle is 4 m above the level of \(V\) (see diagram). The particle and shell rotate with the same constant angular speed. The coefficient of friction between the particle and the shell is \(\mu\).

1. The frictional force on the particle is \(F \mathrm {~N}\), and the normal force of the shell on the particle is \(R \mathrm {~N}\). It is given that the speed of the particle is \(4.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), which is the smallest possible speed for the particle not to slip.
   (a) By resolving vertically, show that \(4 F + 3 R = 19.6\).
   (b) By finding another equation connecting \(F\) and \(R\), find the values of \(F\) and \(R\) and show that \(\mu = 0.336\), correct to 3 significant figures.
2. Find the largest possible angular speed of the shell for which the particle does not slip. \href{http://physicsandmathstutor.com}{physicsandmathstutor.com}
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