Particle on circular wire/arc

A question is this type if and only if a particle moves on a smooth fixed wire in the form of a circular arc in a vertical plane and the question asks when/where it loses contact with the wire.

2 questions · Standard +0.8

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CAIE FP2 2014 June Q4
10 marks Challenging +1.2
4
\includegraphics[max width=\textwidth, alt={}, center]{ab5f2781-e5ce-4fce-bc95-9d7f55ea66d9-2_515_583_1388_781} A smooth wire is in the form of an \(\operatorname { arc } A B\) of a circle, of radius \(a\), that subtends an obtuse angle \(\pi - \theta\) at the centre \(O\) of the circle. It is given that \(\sin \theta = \frac { 1 } { 4 }\). The wire is fixed in a vertical plane, with \(A O\) horizontal and \(B\) below the level of \(O\) (see diagram). A small bead of mass \(m\) is threaded on the wire and projected vertically downwards from \(A\) with speed \(\sqrt { } \left( \frac { 3 } { 10 } g a \right)\).
  1. Find the reaction between the bead and the wire when the bead is vertically below \(O\).
  2. Find the speed of the bead as it leaves the wire at \(B\).
  3. Show that the greatest height reached by the bead is \(\frac { 1 } { 8 } a\) above the level of \(O\).
Edexcel M3 Q3
8 marks Standard +0.3
3. A smooth circular hoop of radius 1 m , with centre \(O\), is fixed in a vertical plane. A small ring \(Q\), of mass 0.1 kg , is threaded onto the hoop and held so that the angle \(Q O H = 30 ^ { \circ }\), where \(H\) is the highest point of the hoop. \(Q\) is released from rest at this position. Find, in terms of \(g\),
  1. the horizontal and vertical components of the acceleration of \(Q\) when it reaches the lowest
    \includegraphics[max width=\textwidth, alt={}, center]{1bcb4e33-b27c-48f0-9540-9ec553e7fe40-1_271_458_620_1528}
    horzonal and graduals moves. Show that the particle starts to move when the extension in the string is 17 cm . point of the hoop;
  2. the magnitude of the reaction between \(Q\) and the hoop at this lowest point. a
    point of the hoop; the magnitude of the reaction between \(Q\) and the hoop at this lowest point.\includegraphics[max width=\textwidth, alt={}]{1bcb4e33-b27c-48f0-9540-9ec553e7fe40-1_46_1257_1469_468} $$\sum \cos + 20 + 2$$ ,
    \includegraphics[max width=\textwidth, alt={}, center]{1bcb4e33-b27c-48f0-9540-9ec553e7fe40-1_90_327_1622_1604}
    \includegraphics[max width=\textwidth, alt={}, center]{1bcb4e33-b27c-48f0-9540-9ec553e7fe40-1_200_81_1512_1850} \section*{MECHANICS 3 (A)TEST PAPER 7 Page 2}