CAIE M1 (Mechanics 1) 2010 November

1 A particle \(P\) is released from rest at a point on a smooth plane inclined at \(30 ^ { \circ }\) to the horizontal. Find the speed of \(P\)

1. when it has travelled 0.9 m ,
2. 0.8 s after it is released.

2
\includegraphics[max width=\textwidth, alt={}, center]{f0200d12-4ab0-4395-804c-e693f7f26507-2_301_1267_616_440} The diagram shows the vertical cross-section \(A B C\) of a fixed surface. \(A B\) is a curve and \(B C\) is a horizontal straight line. The part of the surface containing \(A B\) is smooth and the part containing \(B C\) is rough. \(A\) is at a height of 1.8 m above \(B C\). A particle of mass 0.5 kg is released from rest at \(A\) and travels along the surface to \(C\).

1. Find the speed of the particle at \(B\).
2. Given that the particle reaches \(C\) with a speed of \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), find the work done against the resistance to motion as the particle moves from \(B\) to \(C\).

3
\includegraphics[max width=\textwidth, alt={}, center]{f0200d12-4ab0-4395-804c-e693f7f26507-2_368_853_1503_644} A small smooth pulley is fixed at the highest point \(A\) of a cross-section \(A B C\) of a triangular prism. Angle \(A B C = 90 ^ { \circ }\) and angle \(B C A = 30 ^ { \circ }\). The prism is fixed with the face containing \(B C\) in contact with a horizontal surface. Particles \(P\) and \(Q\) are attached to opposite ends of a light inextensible string, which passes over the pulley. The particles are in equilibrium with \(P\) hanging vertically below the pulley and \(Q\) in contact with \(A C\). The resultant force exerted on the pulley by the string is \(3 \sqrt { } 3 \mathrm {~N}\) (see diagram).

1. Show that the tension in the string is 3 N . The coefficient of friction between \(Q\) and the prism is 0.75 .
2. Given that \(Q\) is in limiting equilibrium and on the point of moving upwards, find its mass.

4 A particle starts from rest at a point \(X\) and moves in a straight line until, 60 seconds later, it reaches a point \(Y\). At time \(t \mathrm {~s}\) after leaving \(X\), the acceleration of the particle is $$\begin{array} { r c c } 0.75 \mathrm {~m} \mathrm {~s} ^ { - 2 } & \text { for } & 0 < t < 4
0 \mathrm {~m} \mathrm {~s} ^ { - 2 } & \text { for } & 4 < t < 54
- 0.5 \mathrm {~m} \mathrm {~s} ^ { - 2 } & \text { for } & 54 < t < 60 \end{array}$$

1. Find the velocity of the particle when \(t = 4\) and when \(t = 60\), and sketch the velocity-time graph.
2. Find the distance \(X Y\).

5 A force of magnitude \(F \mathrm {~N}\) acts in a horizontal plane and has components 27.5 N and - 24 N in the \(x\)-direction and the \(y\)-direction respectively. The force acts at an angle of \(\alpha ^ { \circ }\) below the \(x\)-axis.

1. Find the values of \(F\) and \(\alpha\). A second force, of magnitude 87.6 N , acts in the same plane at \(90 ^ { \circ }\) anticlockwise from the force of magnitude \(F \mathrm {~N}\). The resultant of the two forces has magnitude \(R \mathrm {~N}\) and makes an angle of \(\theta ^ { \circ }\) with the positive \(x\)-axis.
2. Find the values of \(R\) and \(\theta\).

6 A particle travels along a straight line. It starts from rest at a point \(A\) on the line and comes to rest again, 10 seconds later, at another point \(B\) on the line. The velocity \(t\) seconds after leaving \(A\) is $$\begin{array} { r l l } 0.72 t ^ { 2 } - 0.096 t ^ { 3 } & \text { for } & 0 \leqslant t \leqslant 5
2.4 t - 0.24 t ^ { 2 } & \text { for } & 5 \leqslant t \leqslant 10 \end{array}$$

1. Show that there is no instantaneous change in the acceleration of the particle when \(t = 5\).
2. Find the distance \(A B\).

7 A car of mass 1250 kg travels along a horizontal straight road. The power of the car's engine is constant and equal to 24 kW and the resistance to the car's motion is constant and equal to \(R \mathrm {~N}\). The car passes through the point \(A\) on the road with speed \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and acceleration \(0.32 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

1. Find the value of \(R\). The car continues with increasing speed, passing through the point \(B\) on the road with speed \(29.9 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The car subsequently passes through the point \(C\).
2. Find the acceleration of the car at \(B\), giving the answer in \(\mathrm { m } \mathrm { s } ^ { - 2 }\) correct to 3 decimal places.
3. Show that, while the car's speed is increasing, it cannot reach \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
4. Explain why the speed of the car is approximately constant between \(B\) and \(C\).
5. State a value of the approximately constant speed, and the maximum possible error in this value at any point between \(B\) and \(C\). The work done by the car's engine during the motion from \(B\) to \(C\) is 1200 kJ .
6. By assuming the speed of the car is constant from \(B\) to \(C\), find, in either order,
   (a) the approximate time taken for the car to travel from \(B\) to \(C\),
   (b) an approximation for the distance \(B C\). \footnotetext{Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity.
   University of Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. }