CAIE M2 (Mechanics 2) 2017 November

1 A particle \(P\) of mass 0.2 kg is released from rest at a point \(O\) on a smooth horizontal surface. A horizontal force of magnitude \(t \mathrm { e } ^ { - v } \mathrm {~N}\) directed away from \(O\) acts on \(P\), where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the velocity of \(P\) at time \(t \mathrm {~s}\) after release. Find the velocity of \(P\) when \(t = 2\).

2 A uniform solid cone has height 0.6 m and base radius 0.2 m . A uniform hollow cylinder, open at both ends, has the same dimensions. An object is made by putting the cone inside the cylinder so that the base of the cone coincides with one end of the cylinder (see diagram, which shows a cross-section). The total weight of the object is 60 N and its centre of mass is 0.25 m from the base of the cone. Calculate the weight of the cone.

3 A particle \(P\) of mass 0.4 kg is released from rest at a point \(O\) on a smooth plane inclined at \(30 ^ { \circ }\) to the horizontal. \(P\) moves down the line of greatest slope through \(O\). The velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when its displacement from \(O\) is \(x \mathrm {~m}\). A retarding force of magnitude \(0.2 v ^ { 2 } \mathrm {~N}\) acts on \(P\) in the direction \(P O\).

1. Show that \(v \frac { \mathrm {~d} v } { \mathrm {~d} x } = 5 - 0.5 v ^ { 2 }\).
2. Express \(v\) in terms of \(x\).

4
\includegraphics[max width=\textwidth, alt={}, center]{6b220343-1d64-4dbc-a42d-77967eef9c6d-06_264_839_260_653} A light elastic string has natural length 2 m and modulus of elasticity 39 N . The ends of the string are attached to fixed points \(A\) and \(B\) which are at the same horizontal level and 2.4 m apart. A particle \(P\) of mass \(m \mathrm {~kg}\) is attached to the mid-point of the string and hangs in equilibrium at a point 0.5 m below \(A B\) (see diagram).

1. Show that \(m = 0.9\).
   \(P\) is projected vertically downwards from the equilibrium position, and comes to instantaneous rest at a point 1.6 m below \(A B\).
2. Calculate the speed of projection of \(P\).

5
\includegraphics[max width=\textwidth, alt={}, center]{6b220343-1d64-4dbc-a42d-77967eef9c6d-08_449_890_262_630}
\(O A B\) is a uniform lamina in the shape of a quadrant of a circle with centre \(O\) and radius 0.8 m which has its centre of mass at \(G\). The lamina is smoothly hinged at \(A\) to a fixed point and is free to rotate in a vertical plane. A horizontal force of magnitude 12 N acting in the plane of the lamina is applied to the lamina at \(B\). The lamina is in equilibrium with \(A G\) horizontal (see diagram).

1. Calculate the length \(A G\).
2. Find the weight of the lamina.

6 One end of a light elastic string of natural length 0.4 m and modulus of elasticity 8 N is attached to a fixed point \(O\) on a smooth horizontal plane. The other end of the string is attached to a particle \(P\) of mass 0.2 kg which moves on the plane in a circular path with centre \(O\). The speed of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the extension of the string is \(x \mathrm {~m}\).

1. Given that \(v = 2.5\), find \(x\).
   It is given instead that the kinetic energy of \(P\) is twice the elastic potential energy stored in the string.
2. Form two simultaneous equations and hence find \(x\) and \(v\).

7 A small ball \(B\) is projected from a point \(O\) which is \(h \mathrm {~m}\) above a horizontal plane. At time 2 s after projection \(B\) has speed \(18 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and is moving in the direction \(30 ^ { \circ }\) above the horizontal.

1. Find the initial speed and the angle of projection of \(B\).
   \(B\) has speed \(38 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) immediately before it strikes the plane.
2. Calculate \(h\).
   \(B\) bounces when it strikes the plane, and leaves the plane with speed \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) but with its horizontal component of velocity unchanged.
3. Find the total time which elapses between the initial projection of \(B\) and the instant when it strikes the plane for the second time.