A sphere of radius 5.0 cm and mass 1.4 kg is rotating on frictionless bearings around a vertical axis. The sphere’s angu

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A sphere of radius 5.0 cm and mass 1.4 kg is rotating on frictionless bearings around a vertical axis.
The sphere’s angular speed is initially ω = 35 rad s−1 . A finger is applied to the equator of the sphere’s rotation and this generates a constant frictional force and thus a constant torque which brings the sphere to rest in 4.5 s. Assuming that the retarding force acts at 90◦ to a line drawn from the centre of the sphere.
i) Assuming that the initial rotation of the sphere is anticlockwise when viewed from above, describe the direction in which the torque acts. (ii) Calculate the magnitude of the force.
(b) The experiment in part (a) is repeated, using the same frictional force as in (a), but instead of being applied at the equator, the frictional force is now applied higher up, at position A as shown on Figure 1. Will the time taken to bring the sphere to rest be greater, smaller or the same as in part (a)? Explain your answer. (c) The solid sphere is replaced with a hollow sphere of a different material which has the same mass and radius as the solid sphere, but now all of the mass is concentrated in a thin shell. This sphere also initially rotates at ω = 35 rad s−1 and the same frictional force is applied as in part (a), again at the equator of the sphere. Will the time taken to bring the sphere to rest be greater, smaller or the same as for the solid sphere than in part (a)? Explain your answer.
The solid aluminium sphere is removed from its mounting and positioned at rest at the top of a uniform slope. When the sphere is released from rest, it rolls without slipping down the slope. By considering the conservation of energy, show that the speed of the sphere’s centre of mass after it has descended a vertical distance h is given by v = r 10 7 gh.