3 A rough circular hoop, with centre O and radius 1 m , is fixed in a vertical plane. A , B and C are points on the hoop such that A and C are at the same horizontal level as O , and OB makes an angle of \(25 ^ { \circ }\) above the horizontal, as shown in the diagram.
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A bead P of mass 0.3 kg is threaded onto the hoop. P is projected vertically downwards from A on two separate occasions.
- The first time, when P is projected with a speed of \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), it first comes to rest at B .
- The second time, when P is projected with a speed of \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), it first comes to rest at C .
The situation is modelled by assuming that during the motion of P the magnitude of the frictional force exerted by the hoop on P is constant.
- Determine the value of \(v\).
- Comment on the validity of the modelling assumption used in this question.