CAIE M1 (Mechanics 1) 2020 June

Three coplanar forces of magnitudes \(100\text{ N}\), \(50\text{ N}\) and \(50\text{ N}\) act at a point \(A\), as shown in the diagram. The value of \(\cos \alpha\) is \(\frac{4}{5}\). \includegraphics{figure_1} Find the magnitude of the resultant of the three forces and state its direction. [3]

A car of mass \(1800\text{ kg}\) is towing a trailer of mass \(400\text{ kg}\) along a straight horizontal road. The car and trailer are connected by a light rigid tow-bar. The car is accelerating at \(1.5\text{ m s}^{-2}\). There are constant resistance forces of \(250\text{ N}\) on the car and \(100\text{ N}\) on the trailer.

1. Find the tension in the tow-bar. [2]
2. Find the power of the engine of the car at the instant when the speed is \(20\text{ m s}^{-1}\). [3]

A particle \(P\) is projected vertically upwards with speed \(5\text{ m s}^{-1}\) from a point \(A\) which is \(2.8\text{ m}\) above horizontal ground.

1. Find the greatest height above the ground reached by \(P\). [3]
2. Find the length of time for which \(P\) is at a height of more than \(3.6\text{ m}\) above the ground. [4]

The diagram shows a ring of mass \(0.1\text{ kg}\) threaded on a fixed horizontal rod. The rod is rough and the coefficient of friction between the ring and the rod is \(0.8\). A force of magnitude \(T\text{ N}\) acts on the ring in a direction at \(30°\) to the rod, downwards in the vertical plane containing the rod. Initially the ring is at rest. \includegraphics{figure_4}

1. Find the greatest value of \(T\) for which the ring remains at rest. [4]
2. Find the acceleration of the ring when \(T = 3\). [3]

A child of mass \(35\text{ kg}\) is swinging on a rope. The child is modelled as a particle \(P\) and the rope is modelled as a light inextensible string of length \(4\text{ m}\). Initially \(P\) is held at an angle of \(45°\) to the vertical (see diagram). \includegraphics{figure_5}

1. Given that there is no resistance force, find the speed of \(P\) when it has travelled half way along the circular arc from its initial position to its lowest point. [4]
2. It is given instead that there is a resistance force. The work done against the resistance force as \(P\) travels from its initial position to its lowest point is \(X\text{ J}\). The speed of \(P\) at its lowest point is \(4\text{ m s}^{-1}\). Find \(X\). [3]

A particle moves in a straight line \(AB\). The velocity \(v\text{ m s}^{-1}\) of the particle \(t\text{ s}\) after leaving \(A\) is given by \(v = t(5 - 2t)\) where \(k\) is a constant. The displacement of the particle from \(A\), in the direction towards \(B\), is \(2.5\text{ m}\) when \(t = 3\) and is \(2.4\text{ m}\) when \(t = 6\).

1. Find the value of \(k\). Hence find an expression, in terms of \(t\), for the displacement of the particle from \(A\). [7]
2. Find the displacement of the particle from \(A\) when its velocity is a minimum. [4]

A particle \(P\) of mass \(0.3\text{ kg}\), lying on a smooth plane inclined at \(30°\) to the horizontal, is released from rest. \(P\) slides down the plane for a distance of \(2.5\text{ m}\) and then reaches a horizontal plane. There is no change in speed when \(P\) reaches the horizontal plane. A particle \(Q\) of mass \(0.2\text{ kg}\) lies at rest on the horizontal plane \(1.5\text{ m}\) from the end of the inclined plane (see diagram). \(P\) collides directly with \(Q\). \includegraphics{figure_7}

1. It is given that the horizontal plane is smooth and that, after the collision, \(P\) continues moving in the same direction, with speed \(2\text{ m s}^{-1}\). Find the speed of \(Q\) after the collision. [5]
2. It is given instead that the horizontal plane is rough and that when \(P\) and \(Q\) collide, they coalesce and move with speed \(1.2\text{ m s}^{-1}\). Find the coefficient of friction between \(P\) and the horizontal plane. [5]