CAIE M1 (Mechanics 1) 2023 November

1 A block of mass 15 kg slides down a line of greatest slope of an inclined plane. The top of the plane is at a vertical height of 1.6 m above the level of the bottom of the plane. The speed of the block at the top of the plane is \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the speed of the block at the bottom of the plane is \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the work done against the resistance to motion of the block.
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2
\includegraphics[max width=\textwidth, alt={}, center]{308cecda-3bc2-4113-b7dd-ed317c5f32c5-03_638_554_260_792} The diagram shows a smooth ring \(R\), of mass \(m \mathrm {~kg}\), threaded on a light inextensible string. A horizontal force of magnitude 2 N acts on \(R\). The ends of the string are attached to fixed points \(A\) and \(B\) on a vertical wall. The part \(A R\) of the string makes an angle of \(30 ^ { \circ }\) with the vertical, the part \(B R\) makes an angle of \(40 ^ { \circ }\) with the vertical and the string is taut. The ring is in equilibrium. Find the tension in the string and find the value of \(m\).
\includegraphics[max width=\textwidth, alt={}, center]{308cecda-3bc2-4113-b7dd-ed317c5f32c5-04_521_707_259_719} A block of mass 10 kg is at rest on a rough plane inclined at an angle of \(30 ^ { \circ }\) to the horizontal. A force of 120 N is applied to the block at an angle of \(20 ^ { \circ }\) above a line of greatest slope (see diagram). There is a force resisting the motion of the block and 200 J of work is done against this force when the block has moved a distance of 5 m up the plane from rest. Find the speed of the block when it has moved a distance of 5 m up the plane from rest.

4 A particle \(P\) of mass 0.2 kg lies at rest on a rough horizontal plane. A horizontal force of 1.2 N is applied to \(P\).

1. Given that \(P\) is in limiting equilibrium, find the coefficient of friction between \(P\) and the plane.
2. Given instead that the coefficient of friction between \(P\) and the plane is 0.3 , find the distance travelled by \(P\) in the third second of its motion.

5 A particle \(A\) of mass 0.5 kg is projected vertically upwards from horizontal ground with speed \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Find the speed of \(A\) when it reaches a height of 20 m above the ground.
   When \(A\) reaches a height of 20 m , it collides with a particle \(B\) of mass 0.3 kg which is moving downwards in the same vertical line as \(A\) with speed \(32.5 \mathrm {~ms} ^ { - 1 }\). In the collision between the two particles, \(B\) is brought to instantaneous rest.
2. Show that the velocity of \(A\) immediately after the collision is \(4.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) downwards.
3. Find the time interval between \(A\) and \(B\) reaching the ground. You should assume that \(A\) does not bounce when it reaches the ground.

6 A railway engine of mass 120000 kg is towing a coach of mass 60000 kg up a straight track inclined at an angle of \(\alpha\) to the horizontal where \(\sin \alpha = 0.02\). There is a light rigid coupling, parallel to the track, connecting the engine and coach. The driving force produced by the engine is 125000 N and there are constant resistances to motion of 22000 N on the engine and 13000 N on the coach.

1. Find the acceleration of the engine and find the tension in the coupling.
   At an instant when the engine is travelling at \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), it comes to a section of track inclined upwards at an angle \(\beta\) to the horizontal. The power produced by the engine is now 4500000 W and, as a result, the engine maintains a constant speed.
2. Assuming that the resistance forces remain unchanged, find the value of \(\beta\).

7 A particle \(X\) travels in a straight line. The velocity of \(X\) at time \(t\) s after leaving a fixed point \(O\) is denoted by \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where $$v = - 0.1 t ^ { 3 } + 1.8 t ^ { 2 } - 6 t + 5.6$$ The acceleration of \(X\) is zero at \(t = p\) and \(t = q\), where \(p < q\).

1. Find the value of \(p\) and the value of \(q\).
   It is given that the velocity of \(X\) is zero at \(t = 14\).
2. Find the velocities of \(X\) at \(t = p\) and at \(t = q\), and hence sketch the velocity-time graph for the motion of \(X\) for \(0 \leqslant t \leqslant 15\).
3. Find the total distance travelled by \(X\) between \(t = 0\) and \(t = 15\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.