Questions — CAIE M2 (456 questions)

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CAIE M2 2010 June Q5
  1. It is given that when the ball moves with speed \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) the tension in the string \(Q B\) is three times the tension in the string \(P B\). Calculate the radius of the circle. The ball now moves along this circular path with the minimum possible speed.
  2. State the tension in the string \(P B\) in this case, and find the speed of the ball.
CAIE M2 2017 March Q5
  1. Find the tension in the string \(A P\) and the value of \(\omega\).
  2. Find \(m\) and the tension in the string \(B Q\).
    \(6 O\) and \(A\) are fixed points on a rough horizontal surface, with \(O A = 1 \mathrm {~m}\). A particle \(P\) of mass 0.4 kg is projected horizontally with speed \(U \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from \(A\) in the direction \(O A\) and moves in a straight line. After projection, when the displacement of \(P\) from \(O\) is \(x \mathrm {~m}\), the velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The coefficient of friction between the surface and \(P\) is 0.4 . A force of magnitude \(\frac { 0.8 } { x } \mathrm {~N}\) acts on \(P\) in the direction \(P O\).
  3. Show that, while the particle is in motion, \(v \frac { \mathrm {~d} v } { \mathrm {~d} x } = - 4 - \frac { 2 } { x }\).
    It is given that \(P\) comes to instantaneous rest between \(x = 2.0\) and \(x = 2.1\).
  4. Find the set of possible values of \(U\).
CAIE M2 2019 March Q6
  1. Find, in terms of \(r\), the distance of the centre of mass of the prism from the centre of the cylinder.
    \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{b8e52188-f9a6-46fc-90bf-97965c6dd324-11_633_729_258_708} \captionsetup{labelformat=empty} \caption{Fig. 2}
    \end{figure} The prism has weight \(W \mathrm {~N}\) and is placed with its curved surface on a rough horizontal plane. The axis of symmetry of the cross-section makes an angle of \(30 ^ { \circ }\) with the vertical. A horizontal force of magnitude \(P \mathrm {~N}\) acting in the plane of the cross-section through the centre of mass is applied to the cylinder at the highest point of this cross-section (see Fig. 2). The prism rests in limiting equilibrium.
  2. Find the coefficient of friction between the prism and the plane.
CAIE M2 2003 November Q4
  1. Show that the distance of the centre of mass of the lamina from the side \(B C\) is 6.37 cm . \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{be83d46f-bf5b-4382-b424-bb5067626adc-3_671_608_1050_772} \captionsetup{labelformat=empty} \caption{Fig. 2}
    \end{figure} The lamina is smoothly hinged to a wall at \(A\) and is supported, with \(A B\) horizontal, by a light wire attached at \(B\). The other end of the wire is attached to a point on the wall, vertically above \(A\), such that the wire makes an angle of \(30 ^ { \circ }\) with \(A B\) (see Fig. 2). The mass of the lamina is 8 kg . Find
  2. the tension in the wire,
  3. the magnitude of the vertical component of the force acting on the lamina at \(A\).
CAIE M2 2008 November Q4
  1. the base of the cylinder,
  2. the curved surface of the cylinder.
    (ii) \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{5109244c-3062-4f5f-9277-fc6b5b28f2d4-3_348_745_1183_740} \captionsetup{labelformat=empty} \caption{Fig. 2}
    \end{figure} Sphere \(A\) is now attached to one end of a light inextensible string. The string passes through a small smooth hole in the middle of the base of the cylinder. Another small sphere \(B\), of mass 0.25 kg , is attached to the other end of the string. \(B\) hangs in equilibrium below the hole while \(A\) is moving in a horizontal circle of radius 0.2 m (see Fig. 2). Find the angular speed of \(A\).
CAIE M2 2012 November Q4
  1. Find \(r\). The upper cylinder is now fixed to the lower cylinder to create a uniform object.
  2. Show that the centre of mass of the object is $$\frac { 25 h ^ { 2 } + 180 h + 81 } { 50 h + 180 } \mathrm {~m}$$ from \(A\). The object is placed with the plane face containing \(A\) in contact with a rough plane inclined at \(\alpha ^ { \circ }\) to the horizontal, where \(\tan \alpha = 0.5\). The object is on the point of toppling without sliding.
  3. Calculate \(h\).