CAIE M2 (Mechanics 2) 2018 June

2 One end of a light elastic string is attached to a fixed point \(O\). The other end of the string is attached to a particle \(P\) of mass 0.4 kg . The string has natural length 0.6 m and modulus of elasticity 24 N . The particle is released from rest at \(O\). Find the two possible values of the distance \(O P\) for which the particle has speed \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

3
\(A B C\) is an object made from a uniform wire consisting of two straight portions \(A B\) and \(B C\), in which \(A B = a , B C = x\) and angle \(A B C = 90 ^ { \circ }\). When the object is freely suspended from \(A\) and in equilibrium, the angle between \(A B\) and the horizontal is \(\theta\) (see diagram).

1. Show that \(x ^ { 2 } \tan \theta - 2 a x - a ^ { 2 } = 0\).
2. Given that \(\tan \theta = 1.25\), calculate the length of the wire in terms of \(a\).

4 A particle \(P\) is projected from a point \(O\) on horizontal ground with initial speed \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and angle of projection \(30 ^ { \circ }\). At the instant \(t s\) after projection, the horizontal and vertically upwards displacements of \(P\) 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 \(P\).
   \(P\) is at the same height above the ground at two points which are a horizontal distance apart of 15 m .
2. Calculate this height.
   \includegraphics[max width=\textwidth, alt={}, center]{874622ab-4c75-4a32-bae5-eef780ed0cc0-08_607_1022_255_573} A uniform object is made by joining a solid cone of height 0.8 m and base radius 0.6 m and a cylinder. The cylinder has length 0.4 m and radius 0.5 m . The cylinder has a cylindrical hole of length 0.4 m and radius \(x \mathrm {~m}\) drilled through it along the axis of symmetry. A plane face of the cylinder is attached to the base of the cone so that the object has an axis of symmetry perpendicular to its base and passing through the vertex of the cone. The object is placed with points on the base of the cone and the base of the cylinder in contact with a horizontal surface (see diagram). The object is on the point of toppling.
3. Show that the centre of mass of the object is 0.15 m from the base of the cone.
4. Find \(x\).
   [0pt] [The volume of a cone is \(\frac { 1 } { 3 } \pi r ^ { 2 } h\).]

6
\includegraphics[max width=\textwidth, alt={}, center]{874622ab-4c75-4a32-bae5-eef780ed0cc0-10_757_464_258_836} A particle \(P\) of mass 0.2 kg is attached to one end of a light inextensible string of length 0.6 m . The other end of the string is attached to a fixed point \(A\). The particle \(P\) is also attached to one end of a second light inextensible string of length 0.6 m , the other end of which is attached to a fixed point \(B\) vertically below \(A\). The particle moves in a horizontal circle of radius 0.3 m , which has its centre at the mid-point of \(A B\), with both strings straight (see diagram).

1. Calculate the least possible angular speed of \(P\).
   The string \(A P\) will break if its tension exceeds 8 N . The string \(B P\) will break if its tension exceeds 5 N .
2. Find the greatest possible speed of \(P\).
   \(7 \quad\) A particle \(P\) of mass 0.2 kg is released from rest at a point \(O\) above horizontal ground. At time \(t \mathrm {~s}\) after its release the velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) downwards. A vertically downwards force of magnitude \(0.6 t \mathrm {~N}\) acts on \(P\). A vertically upwards force of magnitude \(k \mathrm { e } ^ { - t } \mathrm {~N}\), where \(k\) is a constant, also acts on \(P\).
3. Show that \(\frac { \mathrm { d } v } { \mathrm {~d} t } = 10 - 5 k \mathrm { e } ^ { - t } + 3 t\).
4. Find the greatest value of \(k\) for which \(P\) does not initially move upwards.
5. Given that \(k = 1\), and that \(P\) strikes the ground when \(t = 2\), find the height of \(O\) above the ground. [5]
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.