CAIE M1 (Mechanics 1) 2015 November

1
\includegraphics[max width=\textwidth, alt={}, center]{f23ea8e7-9b81-4192-8c20-8c46aabfecca-2_446_497_258_826} A small ball \(B\) of mass 4 kg is attached to one end of a light inextensible string. A particle \(P\) of mass 3 kg is attached to the other end of the string. The string passes over a fixed smooth pulley. The system is in equilibrium with the string taut and its straight parts vertical. \(B\) is at rest on a rough plane inclined to the horizontal at an angle of \(\alpha\), where \(\cos \alpha = 0.8\) (see diagram). State the tension in the string and find the normal component of the contact force exerted on \(B\) by the plane.

2
\includegraphics[max width=\textwidth, alt={}, center]{f23ea8e7-9b81-4192-8c20-8c46aabfecca-2_195_719_1114_712} A ring of mass 0.2 kg is threaded on a fixed rough horizontal rod and a light inextensible string is attached to the ring at an angle \(\alpha\) above the horizontal, where \(\cos \alpha = 0.96\). The ring is in limiting equilibrium with the tension in the string \(T \mathrm {~N}\) (see diagram). Given that the coefficient of friction between the ring and the rod is 0.25 , find the value of \(T\).
[0pt] [5]

3
\includegraphics[max width=\textwidth, alt={}, center]{f23ea8e7-9b81-4192-8c20-8c46aabfecca-2_296_735_1685_705} Three horizontal forces of magnitudes \(150 \mathrm {~N} , 100 \mathrm {~N}\) and \(P \mathrm {~N}\) have directions as shown in the diagram. The resultant of the three forces is shown by the broken line in the diagram. This resultant has magnitude 120 N and makes an angle \(75 ^ { \circ }\) with the 150 N force. Find the values of \(P\) and \(\theta\).

4
\includegraphics[max width=\textwidth, alt={}, center]{f23ea8e7-9b81-4192-8c20-8c46aabfecca-3_442_495_255_826} Particles \(A\) and \(B\), of masses 0.35 kg and 0.15 kg respectively, are attached to the ends of a light inextensible string which passes over a fixed smooth pulley. The system is at rest with \(B\) held on the horizontal floor, the string taut and its straight parts vertical. \(A\) is at a height of 1.6 m above the floor (see diagram). \(B\) is released and the system begins to move; \(B\) does not reach the pulley. Find

1. the acceleration of the particles and the tension in the string before \(A\) reaches the floor,
2. the greatest height above the floor reached by \(B\).

5 A cyclist and his bicycle have a total mass of 90 kg . The cyclist starts to move with speed \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from the top of a straight hill, of length 500 m , which is inclined at an angle of \(\sin ^ { - 1 } 0.05\) to the horizontal. The cyclist moves with constant acceleration until he reaches the bottom of the hill with speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The cyclist generates 420 W of power while moving down the hill. The resistance to the motion of the cyclist and his bicycle, \(R \mathrm {~N}\), and the cyclist's speed, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), both vary.

1. Show that \(R = \frac { 420 } { v } + 43.56\).
2. Find the cyclist's speed at the mid-point of the hill. Hence find the decrease in the value of \(R\) when the cyclist moves from the top of the hill to the mid-point of the hill, and when the cyclist moves from the mid-point of the hill to the bottom of the hill.

6 A particle \(P\) starts from rest at a point \(O\) of a straight line and moves along the line. The displacement of the particle at time \(t \mathrm {~s}\) after leaving \(O\) is \(x \mathrm {~m}\), where $$x = 0.08 t ^ { 2 } - 0.0002 t ^ { 3 }$$

1. Find the value of \(t\) when \(P\) returns to \(O\) and find the speed of \(P\) as it passes through \(O\) on its return.
2. For the motion of \(P\) until the instant it returns to \(O\), find
   (a) the total distance travelled,
   (b) the average speed.

7 A straight hill \(A B\) has length 400 m with \(A\) at the top and \(B\) at the bottom and is inclined at an angle of \(4 ^ { \circ }\) to the horizontal. A straight horizontal road \(B C\) has length 750 m . A car of mass 1250 kg has a speed of \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at \(A\) when starting to move down the hill. While moving down the hill the resistance to the motion of the car is 2000 N and the driving force is constant. The speed of the car on reaching \(B\) is \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. By using work and energy, find the driving force of the car. On reaching \(B\) the car moves along the road \(B C\). The driving force is constant and twice that when the car was on the hill. The resistance to the motion of the car continues to be 2000 N . Find
2. the acceleration of the car while moving from \(B\) to \(C\),
3. the power of the car's engine as the car reaches \(C\).