CAIE M1 (Mechanics 1) 2015 June

A block is pulled along a horizontal floor by a horizontal rope. The tension in the rope is 500 N and the block moves at a constant speed of \(2.75 \text{ m s}^{-1}\). Find the work done by the tension in 40 s and find the power applied by the tension. [4]

\includegraphics{figure_2} Particles \(A\) and \(B\), of masses 0.35 kg and 0.15 kg respectively, are attached to the ends of a light inextensible string. \(A\) is held at rest on a smooth horizontal surface with the string passing over a small smooth pulley fixed at the edge of the surface. \(B\) hangs vertically below the pulley at a distance \(h\) m above the floor (see diagram). \(A\) is released and the particles move. \(B\) reaches the floor and \(A\) subsequently reaches the pulley with a speed of \(3 \text{ m s}^{-1}\).

1. Explain briefly why the speed with which \(B\) reaches the floor is \(3 \text{ m s}^{-1}\). [1]
2. Find the value of \(h\). [4]

A car of mass 860 kg travels along a straight horizontal road. The power provided by the car's engine is \(P\) W and the resistance to the car's motion is \(R\) N. The car passes through one point with speed \(4.5 \text{ m s}^{-1}\) and acceleration \(4 \text{ m s}^{-2}\). The car passes through another point with speed \(22.5 \text{ m s}^{-1}\) and acceleration \(0.3 \text{ m s}^{-2}\). Find the values of \(P\) and \(R\). [6]

A lorry of mass 12 000 kg moves up a straight hill of length 500 m, starting at the bottom with a speed of \(24 \text{ m s}^{-1}\) and reaching the top with a speed of \(16 \text{ m s}^{-1}\). The top of the hill is 25 m above the level of the bottom of the hill. The resistance to motion of the lorry is 7500 N. Find the driving force of the lorry. [6]

\includegraphics{figure_1} Four coplanar forces of magnitudes 4 N, 8 N, 12 N and 16 N act at a point. The directions in which the forces act are shown in Fig. 1.

1. Find the magnitude and direction of the resultant of the four forces. [5]

\includegraphics{figure_2} The forces of magnitudes 4 N and 16 N exchange their directions and the forces of magnitudes 8 N and 12 N also exchange their directions (see Fig. 2).

1. State the magnitude and direction of the resultant of the four forces in Fig. 2. [2]

A small box of mass 5 kg is pulled at a constant speed of \(2.5 \text{ m s}^{-1}\) down a line of greatest slope of a rough plane inclined at \(10°\) to the horizontal. The pulling force has magnitude 20 N and acts downwards parallel to a line of greatest slope of the plane.

1. Find the coefficient of friction between the box and the plane. [5]

The pulling force is removed while the box is moving at \(2.5 \text{ m s}^{-1}\).

1. Find the distance moved by the box after the instant at which the pulling force is removed. [4]

A particle \(P\) moves on a straight line. It starts at a point \(O\) on the line and returns to \(O\) 100 s later. The velocity of \(P\) is \(v \text{ m s}^{-1}\) at time \(t\) s after leaving \(O\), where $$v = 0.0001t^3 - 0.015t^2 + 0.5t.$$

1. Show that \(P\) is instantaneously at rest when \(t = 0\), \(t = 50\) and \(t = 100\). [2]
2. Find the values of \(v\) at the times for which the acceleration of \(P\) is zero, and sketch the velocity-time graph for \(P\)'s motion for \(0 \leq t \leq 100\). [7]
3. Find the greatest distance of \(P\) from \(O\) for \(0 \leq t \leq 100\). [4]