CAIE M2 (Mechanics 2) 2018 June

A small ball \(B\) is projected from a point \(O\) on horizontal ground. The initial velocity of \(B\) has horizontal and vertically upwards components of \(18 \text{ ms}^{-1}\) and \(25 \text{ ms}^{-1}\) respectively. For the instant \(4 \text{ s}\) after projection, find the speed and direction of motion of \(B\). [4]

\includegraphics{figure_2} A non-uniform rod \(AB\) of length \(0.5 \text{ m}\) and weight \(8 \text{ N}\) is freely hinged to a fixed point at \(A\). The rod makes an angle of \(30°\) with the horizontal with \(B\) above the level of \(A\). The rod is held in equilibrium by a force of magnitude \(12 \text{ N}\) acting in the vertical plane containing the rod at an angle of \(30°\) to \(AB\) applied at \(B\) (see diagram). Find the distance of the centre of mass of the rod from \(A\). [3]

A particle \(P\) of mass \(0.4 \text{ kg}\) is projected horizontally along a smooth horizontal plane from a point \(O\). At time \(t \text{ s}\) after projection the velocity of \(P\) is \(v \text{ ms}^{-1}\). A force of magnitude \(0.8t \text{ N}\) directed away from \(O\) acts on \(P\) and a force of magnitude \(2e^{-t} \text{ N}\) opposes the motion of \(P\).

1. Show that \(\frac{dv}{dt} = 2t - 5e^{-t}\). [2]
2. Given that \(v = 8\) when \(t = 1\), express \(v\) in terms of \(t\). [3]
3. Find the speed of projection of \(P\). [2]

A small object is projected from a point \(O\) with speed \(V \text{ ms}^{-1}\) at an angle of \(45°\) above the horizontal. At time \(t\) after projection, the horizontal and vertically upwards displacements of the object from \(O\) are \(x \text{ m}\) and \(y \text{ m}\) respectively.

1. Express \(x\) and \(y\) in terms of \(t\), and hence find the equation of the path. [4]

The object passes through the point with coordinates \((24, 18)\).

1. Find \(V\). [2]
2. The object passes through two points which are \(22.5 \text{ m}\) above the level of \(O\). Find the values of \(x\) for these points. [3]

A particle \(P\) of mass \(0.7 \text{ kg}\) is attached by a light elastic string to a fixed point \(O\) on a smooth plane inclined at an angle of \(30°\) to the horizontal. The natural length of the string is \(0.5 \text{ m}\) and the modulus of elasticity is \(20 \text{ N}\). The particle \(P\) is projected up the line of greatest slope through \(O\) from a point \(A\) below the level of \(O\). The initial kinetic energy of \(P\) is \(1.8 \text{ J}\) and the initial elastic potential energy in the string is also \(1.8 \text{ J}\).

1. Find the distance \(OA\). [2]
2. Find the greatest speed of \(P\) in the motion. [6]

\includegraphics{figure_6} A particle \(P\) of mass \(0.2 \text{ kg}\) is attached to one end of a light inextensible string of length \(0.6 \text{ m}\). The other end of the string is attached to a particle \(Q\) of mass \(0.3 \text{ kg}\). The string passes through a small hole \(H\) in a smooth horizontal surface. A light elastic string of natural length \(0.3 \text{ m}\) and modulus of elasticity \(15 \text{ N}\) joins \(Q\) to a fixed point \(A\) which is \(0.4 \text{ m}\) vertically below \(H\). The particle \(P\) moves on the surface in a horizontal circle with centre \(H\) (see diagram).

1. Calculate the greatest possible speed of \(P\) for which the elastic string is not extended. [4]
2. Find the distance \(HP\) given that the angular speed of \(P\) is \(8 \text{ rad s}^{-1}\). [5]

\includegraphics{figure_7} A uniform solid cone has height \(1.2 \text{ m}\) and base radius \(0.5 \text{ m}\). A uniform object is made by drilling a cylindrical hole of radius \(0.2 \text{ m}\) through the cone along the axis of symmetry (see diagram).

1. Show that the height of the object is \(0.72 \text{ m}\) and that the volume of the cone removed by the drilling is \(0.0352\pi \text{ m}^3\). [4]

[The volume of a cone is \(\frac{1}{3}\pi r^2 h\).]

1. Find the distance of the centre of mass of the object from its base. [6]