CAIE M2 (Mechanics 2) Specimen

1 A particle is projected with speed \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(50 ^ { \circ }\) above the horizontal. Calculate the time after projection when the particle has speed \(18 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and is rising.

2 One end of a light inextensible string of length 0.5 m is attached to a fixed point \(A\). A particle \(P\) of mass 0.2 kg is attached to the other end of the string. \(P\) moves with constant speed in a horizontal circle with centre \(O\) which is 0.4 m vertically below \(A\).

1. Show that the tension in the string is 2.5 N .
2. Find the speed of \(P\).

3 A particle \(P\) is projected with speed \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(\theta ^ { \circ }\) above the horizontal from a point \(O\) on horizontal ground. At the instant 4 s after projection the particle passes through the point \(A\), where \(O A = 40 \mathrm {~m}\) and the line \(O A\) makes an angle of \(30 ^ { \circ }\) with the horizontal. Calculate \(V\) and \(\theta\).

4
\includegraphics[max width=\textwidth, alt={}, center]{add3948c-3b45-4e67-ac84-e2ca935afd64-05_392_621_255_762} A particle \(P\) of mass 0.4 kg moves with constant speed in a horizontal circle on the smooth inner surface of a fixed hollow hemisphere with centre \(O\) and radius 0.5 m (see diagram).

1. Given that the speed of the particle is \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and its angular speed is \(10 \mathrm { rad } \mathrm { s } ^ { - 1 }\), calculate the angle between \(O P\) and the vertical.
2. Given instead that the magnitude of the force exerted on \(P\) by the hemisphere is 6 N , calculate
   (a) the angle between \(O P\) and the vertical,
   (b) the angular speed of \(P\).

5 A particle \(P\) of mass 0.5 kg is projected vertically upwards from a point on a horizontal surface. A resisting force of magnitude \(0.02 v ^ { 2 } \mathrm {~N}\) acts on \(P\), where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the upward velocity of \(P\) when it is a height of \(x \mathrm {~m}\) above the surface. The initial speed of \(P\) is \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Show that, while \(P\) is moving upwards, \(v \frac { \mathrm {~d} v } { \mathrm {~d} x } = - 10 - 0.04 v ^ { 2 }\).
2. Find the greatest height of \(P\) above the surface.
3. Find the speed of \(P\) immediately before it strikes the surface after descending.

6
\includegraphics[max width=\textwidth, alt={}, center]{add3948c-3b45-4e67-ac84-e2ca935afd64-08_442_953_237_596} An object is formed by joining a hemispherical shell of radius 0.2 m and a solid cone with base radius 0.2 m and height \(h \mathrm {~m}\) along their circumferences. The centre of mass, \(G\), of the object is \(d \mathrm {~m}\) from the vertex of the cone on the axis of symmetry of the object. The object rests in equilibrium on a horizontal plane, with the curved surface of the cone in contact with the plane (see diagram). The object is on the point of toppling.

1. Show that \(d = h + \frac { 0.04 } { h }\).
2. It is given that the cone is uniform and of weight 4 N , and that the hemispherical shell is uniform and of weight \(W \mathrm {~N}\). Given also that \(h = 0.8\), find \(W\).

7 A particle \(P\) of mass \(M \mathrm {~kg}\) is attached to one end of a light elastic string of natural length 0.8 m and modulus of elasticity 12.5 N . The other end of the string is attached to a fixed point \(A\). The particle is released from rest at \(A\) and falls vertically until it comes to instantaneous rest at the point \(B\). The greatest speed of \(P\) during its descent is \(4.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when the extension of the string is \(e \mathrm {~m}\).

1. Show that \(e = 0.64 M\).
2. Find a second equation in \(e\) and \(M\), and hence find \(M\).
3. Calculate the distance \(A B\).