CAIE M1 (Mechanics 1) 2013 June

1 A block is at rest on a rough horizontal plane. The coefficient of friction between the block and the plane is 1.25 .

1. State, giving a reason for your answer, whether the minimum vertical force required to move the block is greater or less than the minimum horizontal force required to move the block. A horizontal force of continuously increasing magnitude \(P \mathrm {~N}\) and fixed direction is applied to the block.
2. Given that the weight of the block is 60 N , find the value of \(P\) when the acceleration of the block is \(4 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

2 A car of mass 1250 kg travels from the bottom to the top of a straight hill of length 600 m , which is inclined at an angle of \(2.5 ^ { \circ }\) to the horizontal. The resistance to motion of the car is constant and equal to 400 N . The work done by the driving force is 450 kJ . The speed of the car at the bottom of the hill is \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the speed of the car at the top of the hill.

3 The top of a cliff is 40 metres above the level of the sea. A man in a boat, close to the bottom of the cliff, is in difficulty and fires a distress signal vertically upwards from sea level. Find

1. the speed of projection of the signal given that it reaches a height of 5 m above the top of the cliff,
2. the length of time for which the signal is above the level of the top of the cliff. The man fires another distress signal vertically upwards from sea level. This signal is above the level of the top of the cliff for \(\sqrt { } ( 17 ) \mathrm { s }\).
3. Find the speed of projection of the second signal.

4 A train of mass 400000 kg is moving on a straight horizontal track. The power of the engine is constant and equal to 1500 kW and the resistance to the train's motion is 30000 N . Find

1. the acceleration of the train when its speed is \(37.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\),
2. the steady speed at which the train can move.

5
\includegraphics[max width=\textwidth, alt={}, center]{2c628138-0729-4160-a95c-d6ab0f199cc5-3_275_663_258_742} A light inextensible string has a particle \(A\) of mass 0.26 kg attached to one end and a particle \(B\) of mass 0.54 kg attached to the other end. The particle \(A\) is held at rest on a rough plane inclined at angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 5 } { 13 }\). The string is taut and parallel to a line of greatest slope of the plane. The string passes over a small smooth pulley at the top of the plane. Particle \(B\) hangs at rest vertically below the pulley (see diagram). The coefficient of friction between \(A\) and the plane is 0.2 . Particle \(A\) is released and the particles start to move.

1. Find the magnitude of the acceleration of the particles and the tension in the string. Particle \(A\) reaches the pulley 0.4 s after starting to move.
2. Find the distance moved by each of the particles.

6
\includegraphics[max width=\textwidth, alt={}, center]{2c628138-0729-4160-a95c-d6ab0f199cc5-3_639_939_1260_603} A particle \(P\) of mass 0.5 kg lies on a smooth horizontal plane. Horizontal forces of magnitudes \(F \mathrm {~N}\), 2.5 N and 2.6 N act on \(P\). The directions of the forces are as shown in the diagram, where \(\tan \alpha = \frac { 12 } { 5 }\) and \(\tan \beta = \frac { 7 } { 24 }\).

1. Given that \(P\) is in equilibrium, find the values of \(F\) and \(\tan \theta\).
2. The force of magnitude \(F \mathrm {~N}\) is removed. Find the magnitude and direction of the acceleration with which \(P\) starts to move.

7 A car driver makes a journey in a straight line from \(A\) to \(B\), starting from rest. The speed of the car increases to a maximum, then decreases until the car is at rest at \(B\). The distance travelled by the car \(t\) seconds after leaving \(A\) is \(0.0000117 \left( 400 t ^ { 3 } - 3 t ^ { 4 } \right)\) metres.

1. Find the distance \(A B\).
2. Find the maximum speed of the car.
3. Find the acceleration of the car
   (a) as it starts from \(A\),
   (b) as it arrives at \(B\).
4. Sketch the velocity-time graph for the journey.