CAIE M1 (Mechanics 1) 2015 June

1 A block \(B\) of mass 2.7 kg is pulled at constant speed along a straight line on a rough horizontal floor. The pulling force has magnitude 25 N and acts at an angle of \(\theta\) above the horizontal. The normal component of the contact force acting on \(B\) has magnitude 20 N .

1. Show that \(\sin \theta = 0.28\).
2. Find the work done by the pulling force in moving the block a distance of 5 m .

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\includegraphics[max width=\textwidth, alt={}, center]{f4f2996b-5382-4b0d-9804-b5f5945946b3-2_636_519_664_813} Three horizontal forces of magnitudes \(F \mathrm {~N} , 63 \mathrm {~N}\) and 25 N act at \(O\), the origin of the \(x\)-axis and \(y\)-axis. The forces are in equilibrium. The force of magnitude \(F \mathrm {~N}\) makes an angle \(\theta\) anticlockwise with the positive \(x\)-axis. The force of magnitude 63 N acts along the negative \(y\)-axis. The force of magnitude 25 N acts at \(\tan ^ { - 1 } 0.75\) clockwise from the negative \(x\)-axis (see diagram). Find the value of \(F\) and the value of \(\tan \theta\).

3 A block of weight 6.1 N slides down a slope inclined at \(\tan ^ { - 1 } \left( \frac { 11 } { 60 } \right)\) to the horizontal. The coefficient of friction between the block and the slope is \(\frac { 1 } { 4 }\). The block passes through a point \(A\) with speed \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find how far the block moves from \(A\) before it comes to rest.

4 A lorry of mass 14000 kg moves along a road starting from rest at a point \(O\). It reaches a point \(A\), and then continues to a point \(B\) which it reaches with a speed of \(24 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The part \(O A\) of the road is straight and horizontal and has length 400 m . The part \(A B\) of the road is straight and is inclined downwards at an angle of \(\theta ^ { \circ }\) to the horizontal and has length 300 m .

1. For the motion from \(O\) to \(B\), find the gain in kinetic energy of the lorry and express its loss in potential energy in terms of \(\theta\). The resistance to the motion of the lorry is 4800 N and the work done by the driving force of the lorry from \(O\) to \(B\) is 5000 kJ .
2. Find the value of \(\theta\).

5 A cyclist and her bicycle have a total mass of 84 kg . She works at a constant rate of \(P \mathrm {~W}\) while moving on a straight road which is inclined to the horizontal at an angle \(\theta\), where \(\sin \theta = 0.1\). When moving uphill, the cyclist's acceleration is \(1.25 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) at an instant when her speed is \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). When moving downhill, the cyclist's acceleration is \(1.25 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) at an instant when her speed is \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The resistance to the cyclist's motion, whether the cyclist is moving uphill or downhill, is \(R \mathrm {~N}\). Find the values of \(P\) and \(R\).

6 Two particles \(A\) and \(B\) start to move at the same instant from a point \(O\). The particles move in the same direction along the same straight line. The acceleration of \(A\) at time \(t \mathrm {~s}\) after starting to move is \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\), where \(a = 0.05 - 0.0002 t\).

1. Find A's velocity when \(t = 200\) and when \(t = 500\).
   \(B\) moves with constant acceleration for the first 200 s and has the same velocity as \(A\) when \(t = 200 . B\) moves with constant retardation from \(t = 200\) to \(t = 500\) and has the same velocity as \(A\) when \(t = 500\).
2. Find the distance between \(A\) and \(B\) when \(t = 500\).

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\includegraphics[max width=\textwidth, alt={}, center]{f4f2996b-5382-4b0d-9804-b5f5945946b3-3_376_1052_1171_548} Particles \(A\) and \(B\), of masses 0.3 kg and 0.7 kg respectively, are attached to the ends of a light inextensible string. Particle \(A\) is held at rest on a rough horizontal table with the string passing over a smooth pulley fixed at the edge of the table. The coefficient of friction between \(A\) and the table is 0.2 . Particle \(B\) hangs vertically below the pulley at a height of 0.5 m above the floor (see diagram). The system is released from rest and 0.25 s later the string breaks. A does not reach the pulley in the subsequent motion. Find

1. the speed of \(B\) immediately before it hits the floor,
2. the total distance travelled by \(A\).