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\includegraphics[max width=\textwidth, alt={}, center]{98bbefd8-b3dd-49f1-8591-e939282cb05c-2_170_616_1649_767} A small sphere \(S\) of mass \(m \mathrm {~kg}\) is moving inside a fixed smooth hollow cylinder whose axis is vertical. \(S\) moves with constant speed in a horizontal circle of radius 0.4 m and is in contact with both the plane base and the curved surface of the cylinder (see diagram).

1. Given that the horizontal and vertical forces exerted on \(S\) by the cylinder have equal magnitudes, calculate the speed of \(S\).
   \(S\) is now attached to the centre of the base of the cylinder by a horizontal light elastic string of natural length 0.25 m and modulus of elasticity 13 N . The sphere \(S\) is set in motion and moves in a horizontal circle with constant angular speed \(\omega \mathrm { rads } ^ { - 1 }\) and is in contact with both the plane base and the curved surface of the cylinder.
2. It is given that the magnitudes of the horizontal and vertical forces exerted on \(S\) by the cylinder are equal if \(\omega = 8\). Calculate \(m\).
3. For the value of \(m\) found in part (ii), find the least possible value of \(\omega\) for the motion.