CAIE M2 (Mechanics 2) 2017 June

1 A particle is projected with speed \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(60 ^ { \circ }\) above the horizontal. Calculate the time after projection when the particle is descending at an angle of \(40 ^ { \circ }\) below the horizontal.
\begin{figure}[h] \end{figure} One end of a light inextensible string is attached to a fixed point \(A\). The other end of the string is attached to a particle \(P\) of mass \(m \mathrm {~kg}\) which hangs vertically below \(A\). The particle is also attached to one end of a light elastic string of natural length 0.25 m . The other end of this string is attached to a point \(B\) which is 0.6 m from \(P\) and on the same horizontal level as \(P\). Equilibrium is maintained by a horizontal force of magnitude 7 N applied to \(P\) (see Fig. 1).

1. Calculate the modulus of elasticity of the elastic string.
   \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{7800deca-98e8-4eb4-9176-288bb1f44fec-05_348_488_262_826} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure} \(P\) is released from rest by removing the 7 N force. In its subsequent motion \(P\) first comes to instantaneous rest at a point where \(B P = 0.3 \mathrm {~m}\) and the elastic string makes an angle of \(30 ^ { \circ }\) with the horizontal (see Fig. 2).
2. Find the value of \(m\).

3
\includegraphics[max width=\textwidth, alt={}, center]{7800deca-98e8-4eb4-9176-288bb1f44fec-06_351_607_269_769} An object is made from a uniform solid hemisphere of radius 0.56 m and centre \(O\) by removing a hemisphere of radius 0.28 m and centre \(O\). The diagram shows a cross-section through \(O\) of the object.

1. Calculate the distance of the centre of mass of the object from \(O\).
   [0pt] [The volume of a hemisphere is \(\frac { 2 } { 3 } \pi r ^ { 3 }\).]
   The object has weight 24 N . A uniform hemisphere \(H\) of radius 0.28 m is placed in the hollow part of the object to create a non-uniform hemisphere with centre \(O\). The centre of mass of the non-uniform hemisphere is 0.15 m from \(O\).
2. Calculate the weight of \(H\).

4 A particle is projected from a point \(O\) on horizontal ground. The initial components of the velocity of the particle are \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) horizontally and \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) vertically. At time \(t \mathrm {~s}\) after projection, the horizontal and vertically upwards displacements of the particle from \(O\) are \(x \mathrm {~m}\) and \(y \mathrm {~m}\) respectively.

1. Express \(x\) and \(y\) in terms of \(t\), and hence find the equation of the trajectory of the particle.
   \includegraphics[max width=\textwidth, alt={}, center]{7800deca-98e8-4eb4-9176-288bb1f44fec-08_63_1569_488_328}
   The horizontal ground is at the top of a vertical cliff. The point \(O\) is at a distance \(d \mathrm {~m}\) from the edge of the cliff. The particle is projected towards the edge of the cliff and does not strike the ground before it passes over the edge of the cliff.
2. Show that \(d\) is less than 30 .
3. Find the value of \(x\) when the particle is 14 m below the level of \(O\).
   \includegraphics[max width=\textwidth, alt={}, center]{7800deca-98e8-4eb4-9176-288bb1f44fec-10_501_614_258_762} A uniform semicircular lamina of radius 0.7 m and weight 14 N has diameter \(A B\). The lamina is in a vertical plane with \(A\) freely pivoted at a fixed point. The straight edge \(A B\) rests against a small smooth peg \(P\) above the level of \(A\). The angle between \(A B\) and the horizontal is \(30 ^ { \circ }\) and \(A P = 0.9 \mathrm {~m}\) (see diagram).
4. Show that the magnitude of the force exerted by the peg on the lamina is 7.12 N , correct to 3 significant figures.
5. Find the angle with the horizontal of the force exerted by the pivot on the lamina at \(A\).

6 A particle \(P\) of mass 0.15 kg is attached to one end of a light elastic string of natural length 0.4 m and modulus of elasticity 12 N . The other end of the string is attached to a fixed point \(A\). The particle \(P\) moves in a horizontal circle which has its centre vertically below \(A\), with the string inclined at \(\theta ^ { \circ }\) to the vertical and \(A P = 0.5 \mathrm {~m}\).

1. Find the angular speed of \(P\) and the value of \(\theta\).
2. Calculate the difference between the elastic potential energy stored in the string and the kinetic energy of \(P\).
   \(7 \quad\) A particle \(P\) of mass 0.5 kg is at rest at a point \(O\) on a rough horizontal surface. At time \(t = 0\), where \(t\) is in seconds, a horizontal force acting in a fixed direction is applied to \(P\). At time \(t \mathrm {~s}\) the magnitude of the force is \(0.6 t ^ { 2 } \mathrm {~N}\) and the velocity of \(P\) away from \(O\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). It is given that \(P\) remains at rest at \(O\) until \(t = 0.5\).
3. Calculate the coefficient of friction between \(P\) and the surface, and show that $$\frac { \mathrm { d } v } { \mathrm {~d} t } = 1.2 t ^ { 2 } - 0.3 \quad \text { for } t > 0.5$$
4. Express \(v\) in terms of \(t\) for \(t > 0.5\).
5. Find the displacement of \(P\) from \(O\) when \(t = 1.2\).