CAIE M1 (Mechanics 1) 2019 November

A crane is lifting a load of 1250 kg vertically at a constant speed \(V\) m s\(^{-1}\). Given that the power of the crane is a constant 20 kW, find the value of \(V\). [2]

The total mass of a cyclist and her bicycle is 75 kg. The cyclist ascends a straight hill of length 0.7 km inclined at 1.5° to the horizontal. Her speed at the bottom of the hill is 10 m s\(^{-1}\) and at the top it is 5 m s\(^{-1}\). There is a resistance to motion, and the work done against this resistance as the cyclist ascends the hill is 2000 J. The cyclist exerts a constant force of magnitude \(F\) N in the direction of motion. Find \(F\). [5]

A block of mass 3 kg is at rest on a rough plane inclined at 60° to the horizontal. A force of magnitude 15 N acting up a line of greatest slope of the plane is just sufficient to prevent the block from sliding down the plane.

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

The force of magnitude 15 N is now replaced by a force of magnitude \(X\) N acting up the line of greatest slope.

1. Find the greatest value of \(X\) for which the block does not move. [2]

\includegraphics{figure_4} Two blocks \(A\) and \(B\) of masses 4 kg and 5 kg respectively are joined by a light inextensible string. The blocks rest on a smooth plane inclined at an angle \(\alpha\) to the horizontal, where tan \(\alpha = \frac{7}{24}\). The string is parallel to a line of greatest slope of the plane with \(B\) above \(A\). A force of magnitude 36 N acts on \(B\), parallel to a line of greatest slope of the plane (see diagram).

1. Find the acceleration of the blocks and the tension in the string. [5]

1. At a particular instant, the speed of the blocks is 1 m s\(^{-1}\). Find the time, after this instant, that it takes for the blocks to travel 0.65 m. [2]

\includegraphics{figure_5} A small ring \(P\) is threaded on a fixed smooth horizontal rod \(AB\). Three horizontal forces of magnitudes 4.5 N, 7.5 N and \(F\) N act on \(P\) (see diagram).

1. Given that these three forces are in equilibrium, find the values of \(F\) and \(\theta\). [6]

1. It is given instead that the values of \(F\) and \(\theta\) are 9.5 and 30 respectively, and the acceleration of the ring is 1.5 m s\(^{-2}\). Find the mass of the ring. [2]

A particle of mass 0.4 kg is released from rest at a height of 1.8 m above the surface of the water in a tank. There is no instantaneous change of speed when the particle enters the water. The water exerts an upward force of 5.6 N on the particle when it is in the water.

1. Find the velocity of the particle at the instant when it reaches the surface of the water. [2]

1. Find the time that it takes from the instant when the particle enters the water until it comes to instantaneous rest in the water. You may assume that the tank is deep enough so that the particle does not reach the bottom of the tank. [4]

1. Sketch a velocity-time graph for the motion of the particle from the instant at which it is released until it comes to instantaneous rest in the water. [3]

A particle moves in a straight line, starting from rest at a point \(O\), and comes to instantaneous rest at a point \(P\). The velocity of the particle at time \(t\) s after leaving \(O\) is \(v\) m s\(^{-1}\), where $$v = 0.6t^2 - 0.12t^3.$$

1. Show that the distance \(OP\) is 6.25 m. [5]

On another occasion, the particle also moves in the same straight line. On this occasion, the displacement of the particle at time \(t\) s after leaving \(O\) is \(s\) m, where $$s = kt^3 + ct^5.$$ It is given that the particle passes point \(P\) with velocity 1.25 m s\(^{-1}\) at time \(t = 5\).

1. Find the values of the constants \(k\) and \(c\). [5]

1. Find the acceleration of the particle at time \(t = 5\). [2]