CAIE M1 (Mechanics 1) 2017 June

One end of a light inextensible string is attached to a block. The string makes an angle of \(\theta°\) with the horizontal. The tension in the string is \(20\) N. The string pulls the block along a horizontal surface at a constant speed of \(1.5\) m s\(^{-1}\) for \(12\) s. The work done by the tension in the string is \(50\) J. Find \(\theta\). [3]

\includegraphics{figure_2} The diagram shows a wire \(ABCD\) consisting of a straight part \(AB\) of length \(5\) m and a part \(BCD\) in the shape of a semicircle of radius \(6\) m and centre \(O\). The diameter \(BD\) of the semicircle is horizontal and \(AB\) is vertical. A small ring is threaded onto the wire and slides along the wire. The ring starts from rest at \(A\). The part \(AB\) of the wire is rough, and the ring accelerates at a constant rate of \(2.5\) m s\(^{-2}\) between \(A\) and \(B\).

1. Show that the speed of the ring as it reaches \(B\) is \(5\) m s\(^{-1}\). [1]

A particle \(A\) moves in a straight line with constant speed \(10\) m s\(^{-1}\). Two seconds after \(A\) passes a point \(O\) on the line, a particle \(B\) passes through \(O\), moving along the line in the same direction as \(A\). Particle \(B\) has speed \(16\) m s\(^{-1}\) at \(O\) and has a constant deceleration of \(2\) m s\(^{-2}\).

1. Find expressions, in terms of \(t\), for the displacement from \(O\) of each particle \(t\) s after \(B\) passes through \(O\). [3]
2. Find the distance between the particles when \(B\) comes to instantaneous rest. [3]
3. Find the minimum distance between the particles. [3]

A car of mass \(1200\) kg is moving on a straight road against a constant force of \(850\) N resisting the motion.

1. On a part of the road that is horizontal, the car moves with a constant speed of \(42\) m s\(^{-1}\).
   1. Calculate, in kW, the power developed by the engine of the car. [2]
   2. Given that this power is suddenly increased by \(6\) kW, find the instantaneous acceleration of the car. [3]
2. On a part of the road that is inclined at \(\theta°\) to the horizontal, the car moves up the hill at a constant speed of \(24\) m s\(^{-1}\), with the engine working at \(80\) kW. Find \(\theta\). [4]

\includegraphics{figure_5} A particle of mass \(0.12\) kg is placed on a plane which is inclined at an angle of \(40°\) to the horizontal. The particle is kept in equilibrium by a force of magnitude \(P\) N acting up the plane at an angle of \(30°\) above a line of greatest slope, as shown in the diagram. The coefficient of friction between the particle and the plane is \(0.32\). Find the set of possible values of \(P\). [8]

\includegraphics{figure_6} The diagram shows a fixed block with a horizontal top surface and a surface which is inclined at an angle of \(\theta°\) to the horizontal, where \(\sin \theta = \frac{3}{5}\). A particle \(A\) of mass \(0.3\) kg rests on the horizontal surface and is attached to one end of a light inextensible string. The string passes over a small smooth pulley \(P\) fixed at the edge of the block. The other end of the string is attached to a particle \(B\) of mass \(1.5\) kg which rests on the sloping surface of the block. The system is released from rest with the string taut.

1. Given that the block is smooth, find the acceleration of particle \(A\) and the tension in the string. [5]
2. It is given instead that the block is rough. The coefficient of friction between \(A\) and the block is \(\mu\) and the coefficient of friction between \(B\) and the block is also \(\mu\). In the first \(3\) seconds of the motion, \(A\) does not reach \(P\) and \(B\) does not reach the bottom of the sloping surface. The speed of the particles after \(3\) s is \(5\) m s\(^{-1}\). Find the acceleration of particle \(A\) and the value of \(\mu\). [9]