CAIE M1 (Mechanics 1) 2012 November

1
\includegraphics[max width=\textwidth, alt={}, center]{631ddcd9-17c0-4a15-8671-40788c3a84d3-2_366_780_251_680}
\(A B C D\) is a semi-circular cross-section, in a vertical plane, of the inner surface of half a hollow cylinder of radius 2.5 m which is fixed with its axis horizontal. \(A D\) is horizontal, \(B\) is the lowest point of the cross-section and \(C\) is at a height of 1.8 m above the level of \(B\) (see diagram). A particle \(P\) of mass 0.8 kg is released from rest at \(A\) and comes to instantaneous rest at \(C\).

1. Find the work done on \(P\) by the resistance to motion while \(P\) travels from \(A\) to \(C\). The work done on \(P\) by the resistance to motion while \(P\) travels from \(A\) to \(B\) is 0.6 times the work done while \(P\) travels from \(A\) to \(C\).
2. Find the speed of \(P\) when it passes through \(B\).

2 A particle moves in a straight line. Its velocity \(t\) seconds after leaving a fixed point \(O\) on the line is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where \(v = 0.2 t + 0.006 t ^ { 2 }\). For the instant when the acceleration of the particle is 2.5 times its initial acceleration,

1. show that \(t = 25\),
2. find the displacement of the particle from \(O\).

3 A particle \(P\) is projected vertically upwards, from a point \(O\), with a velocity of \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The point \(A\) is the highest point reached by \(P\). Find

1. the speed of \(P\) when it is at the mid-point of \(O A\),
2. the time taken for \(P\) to reach the mid-point of \(O A\) while moving upwards.

4
\includegraphics[max width=\textwidth, alt={}, center]{631ddcd9-17c0-4a15-8671-40788c3a84d3-2_396_880_1996_630} A particle \(P\) of weight 21 N is attached to one end of each of two light inextensible strings, \(S _ { 1 }\) and \(S _ { 2 }\), of lengths 0.52 m and 0.25 m respectively. The other end of \(S _ { 1 }\) is attached to a fixed point \(A\), and the other end of \(S _ { 2 }\) is attached to a fixed point \(B\) at the same horizontal level as \(A\). The particle \(P\) hangs in equilibrium at a point 0.2 m below the level of \(A B\) with both strings taut (see diagram). Find the tension in \(S _ { 1 }\) and the tension in \(S _ { 2 }\).

5 An object of mass 12 kg slides down a line of greatest slope of a smooth plane inclined at \(10 ^ { \circ }\) to the horizontal. The object passes through points \(A\) and \(B\) with speeds \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively.

1. Find the increase in kinetic energy of the object as it moves from \(A\) to \(B\).
2. Hence find the distance \(A B\), assuming there is no resisting force acting on the object. The object is now pushed up the plane from \(B\) to \(A\), with constant speed, by a horizontal force.
3. Find the magnitude of this force.

6
\includegraphics[max width=\textwidth, alt={}, center]{631ddcd9-17c0-4a15-8671-40788c3a84d3-3_255_511_794_817} The diagram shows a particle of mass 0.6 kg on a plane inclined at \(25 ^ { \circ }\) to the horizontal. The particle is acted on by a force of magnitude \(P \mathrm {~N}\) directed up the plane parallel to a line of greatest slope. The coefficient of friction between the particle and the plane is 0.36 . Given that the particle is in equilibrium, find the set of possible values of \(P\).

7
\includegraphics[max width=\textwidth, alt={}, center]{631ddcd9-17c0-4a15-8671-40788c3a84d3-3_565_828_1402_660} Particles \(A\) and \(B\) have masses 0.32 kg and 0.48 kg respectively. The particles are attached to the ends of a light inextensible string which passes over a small smooth pulley fixed at the edge of a smooth horizontal table. Particle \(B\) is held at rest on the table at a distance of 1.4 m from the pulley. \(A\) hangs vertically below the pulley at a height of 0.98 m above the floor (see diagram). \(A , B\), the string and the pulley are all in the same vertical plane. \(B\) is released and \(A\) moves downwards.

1. Find the acceleration of \(A\) and the tension in the string.
   \(A\) hits the floor and \(B\) continues to move towards the pulley. Find the time taken, from the instant that \(B\) is released, for
2. \(A\) to reach the floor,
3. \(B\) to reach the pulley.