CAIE M1 (Mechanics 1) 2019 March

1
\includegraphics[max width=\textwidth, alt={}, center]{06df8c0d-dd38-4e3b-b1a4-72120a81050e-02_415_623_255_760} A small ring \(P\) of mass 0.03 kg is threaded on a rough vertical rod. A light inextensible string is attached to the ring and is pulled upwards at an angle of \(15 ^ { \circ }\) to the horizontal. The tension in the string is 2.5 N (see diagram). The ring is in limiting equilibrium and on the point of sliding up the rod. Find the coefficient of friction between the ring and the rod.

2 A particle is projected vertically upwards with speed \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point on horizontal ground.

1. Show that the maximum height above the ground reached by the particle is 45 m .
2. Find the time that it takes for the particle to reach a height of 33.75 m above the ground for the first time. Find also the speed of the particle at this time.
   \includegraphics[max width=\textwidth, alt={}, center]{06df8c0d-dd38-4e3b-b1a4-72120a81050e-04_645_661_260_740} Four coplanar forces of magnitudes \(F \mathrm {~N} , 5 \mathrm {~N} , 25 \mathrm {~N}\) and 15 N are acting at a point \(P\) in the directions shown in the diagram. Given that the forces are in equilibrium, find the values of \(F\) and \(\alpha\).

4 A car of mass 1500 kg is pulling a trailer of mass 300 kg along a straight horizontal road at a constant speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The system of the car and trailer is modelled as two particles, connected by a light rigid horizontal rod. The power of the car's engine is 6000 W . There are constant resistances to motion of \(R \mathrm {~N}\) on the car and 80 N on the trailer.

1. Find the value of \(R\).
   The power of the car's engine is increased to 12500 W . The resistance forces do not change.
2. Find the acceleration of the car and trailer and the tension in the rod at an instant when the speed of the car is \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

5
\includegraphics[max width=\textwidth, alt={}, center]{06df8c0d-dd38-4e3b-b1a4-72120a81050e-08_759_1447_260_349} The velocity of a particle moving in a straight line is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at time \(t\) seconds after leaving a fixed point \(O\). The diagram shows a velocity-time graph which models the motion of the particle from \(t = 0\) to \(t = 16\). The graph consists of five straight line segments. The acceleration of the particle from \(t = 0\) to \(t = 3\) is \(3 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). The velocity of the particle at \(t = 5\) is \(7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and it comes to instantaneous rest at \(t = 8\). The particle then comes to rest again at \(t = 16\). The minimum velocity of the particle is \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Find the distance travelled by the particle in the first 8 s of its motion.
2. Given that when the particle comes to rest at \(t = 16\) its displacement from \(O\) is 32 m , find the value of \(V\).

6 A particle moves in a straight line. It starts from rest at a fixed point \(O\) on the line. Its acceleration at time \(t \mathrm {~s}\) after leaving \(O\) is \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\), where \(a = 0.4 t ^ { 3 } - 4.8 t ^ { \frac { 1 } { 2 } }\).

1. Show that, in the subsequent motion, the acceleration of the particle when it comes to instantaneous rest is \(16 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
2. Find the displacement of the particle from \(O\) at \(t = 5\).
   \includegraphics[max width=\textwidth, alt={}, center]{06df8c0d-dd38-4e3b-b1a4-72120a81050e-12_554_878_260_635} The diagram shows the vertical cross-section \(P Q R\) of a slide. The part \(P Q\) is a straight line of length 8 m inclined at angle \(\alpha\) to the horizontal, where \(\sin \alpha = 0.8\). The straight part \(P Q\) is tangential to the curved part \(Q R\), and \(R\) is \(h \mathrm {~m}\) above the level of \(P\). The straight part \(P Q\) of the slide is rough and the curved part \(Q R\) is smooth. A particle of mass 0.25 kg is projected with speed \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from \(P\) towards \(Q\) and comes to rest at \(R\). The coefficient of friction between the particle and \(P Q\) is 0.5 .
3. Find the work done by the friction force during the motion of the particle from \(P\) to \(Q\).
4. Hence find the speed of the particle at \(Q\).
5. Find the value of \(h\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.