CAIE M2 (Mechanics 2) 2014 November

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\includegraphics[max width=\textwidth, alt={}, center]{84a2b2eb-a750-4047-864b-4a165fc66b2a-2_812_1218_479_468} A uniform solid cone with height 0.8 m and semi-vertical angle \(30 ^ { \circ }\) has weight 20 N . The cone rests in equilibrium with a single point \(P\) of its base in contact with a rough horizontal surface, and its vertex \(V\) vertically above \(P\). Equilibrium is maintained by a force of magnitude \(F \mathrm {~N}\) acting along the axis of symmetry of the cone and applied to \(V\) (see diagram).

1. Show that the moment of the weight of the cone about \(P\) is 6 Nm .
2. Hence find \(F\).

3 One end of a light elastic string of natural length 1.6 m and modulus of elasticity 28 N is attached to a fixed point \(O\). The other end of the string is attached to a particle \(P\) of mass 0.35 kg which hangs in equilibrium vertically below \(O\). The particle \(P\) is projected vertically upwards from the equilibrium position with speed \(1.8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Calculate the speed of \(P\) at the instant the string first becomes slack.

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\includegraphics[max width=\textwidth, alt={}, center]{84a2b2eb-a750-4047-864b-4a165fc66b2a-3_611_977_260_584}
\(A B C D E F\) is the cross-section through the centre of mass of a uniform solid prism. \(A B C F\) is a rectangle in which \(A B = C F = 1.6 \mathrm {~m}\), and \(B C = A F = 0.4 \mathrm {~m}\). \(C D E\) is a triangle in which \(C D = 1.8 \mathrm {~m}\), \(C E = 0.4 \mathrm {~m}\), and angle \(D C E = 90 ^ { \circ }\). The prism stands on a rough horizontal surface. A horizontal force of magnitude \(T \mathrm {~N}\) acts at \(B\) in the direction \(C B\) (see diagram). The prism is in equilibrium.

1. Show that the distance of the centre of mass of the prism from \(A B\) is 0.488 m .
2. Given that the weight of the prism is 100 N , find the greatest and least possible values of \(T\).

5 The equation of the trajectory of a small ball \(B\) projected from a fixed point \(O\) is $$y = - 0.05 x ^ { 2 }$$ where \(x\) and \(y\) are, respectively, the displacements in metres of \(B\) from \(O\) in the horizontal and vertically upwards directions.

1. Show that \(B\) is projected horizontally, and find its speed of projection.
2. Find the value of \(y\) when the direction of motion of \(B\) is \(60 ^ { \circ }\) below the horizontal, and find the corresponding speed of \(B\).
   \(6 \quad O , A\) and \(B\) are three points in a straight line on a smooth horizontal surface. A particle \(P\) of mass 0.6 kg moves along the line. At time \(t \mathrm {~s}\) the particle has displacement \(x \mathrm {~m}\) from \(O\) and speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The only horizontal force acting on \(P\) has magnitude \(0.4 v ^ { \frac { 1 } { 2 } } \mathrm {~N}\) and acts in the direction \(O A\). Initially the particle is at \(A\), where \(x = 1\) and \(v = 1\).
3. Show that \(3 v ^ { \frac { 1 } { 2 } } \frac { \mathrm {~d} v } { \mathrm {~d} x } = 2\).
4. Express \(v\) in terms of \(x\).
5. Given that \(A B = 7 \mathrm {~m}\), find the value of \(t\) when \(P\) passes through \(B\).

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\includegraphics[max width=\textwidth, alt={}, center]{84a2b2eb-a750-4047-864b-4a165fc66b2a-4_558_857_260_644} One end of a light elastic string with modulus of elasticity 15 N is attached to a fixed point \(A\) which is 2 m vertically above a fixed small smooth ring \(R\). The string has natural length 2 m and it passes through \(R\). The other end of the string is attached to a particle \(P\) of mass \(m \mathrm {~kg}\) which moves with constant angular speed \(\omega \mathrm { rad } \mathrm { s } ^ { - 1 }\) in a horizontal circle which has its centre 0.4 m vertically below the ring. \(P R\) makes an acute angle \(\theta\) with the vertical (see diagram).

1. Show that the tension in the string is \(\frac { 3 } { \cos \theta } \mathrm {~N}\) and hence find the value of \(m\).
2. Show that the value of \(\omega\) does not depend on \(\theta\). It is given that for one value of \(\theta\) the elastic potential energy stored in the string is twice the kinetic energy of \(P\).
3. Find this value of \(\theta\).