CAIE M1 (Mechanics 1) 2016 November

1
\includegraphics[max width=\textwidth, alt={}, center]{a92f97e2-343f-4cac-ae38-f18a4ad49055-2_241_823_264_660} Two particles \(P\) and \(Q\), of masses 0.6 kg and 0.4 kg respectively, are connected by a light inextensible string. The string passes over a small smooth light pulley fixed at the edge of a smooth horizontal table. Initially \(P\) is held at rest on the table and \(Q\) hangs vertically (see diagram). \(P\) is then released. Find the tension in the string and the acceleration of \(Q\).

2 A particle of mass 0.1 kg is released from rest on a rough plane inclined at \(20 ^ { \circ }\) to the horizontal. It is given that, 5 seconds after release, the particle has a speed of \(2 \mathrm {~ms} ^ { - 1 }\).

1. Find the acceleration of the particle and hence show that the magnitude of the frictional force acting on the particle is 0.302 N , correct to 3 significant figures.
2. Find the coefficient of friction between the particle and the plane.

3 A particle \(P\) is projected vertically upwards from a point \(O\). When the particle is at a height of 0.5 m , its speed is \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find

1. the greatest height reached by the particle above \(O\),
2. the time after projection at which the particle returns to \(O\).

4
\includegraphics[max width=\textwidth, alt={}, center]{a92f97e2-343f-4cac-ae38-f18a4ad49055-2_334_832_1617_660} Three coplanar forces of magnitudes \(F \mathrm {~N} , 2 F \mathrm {~N}\) and 15 N act at a point \(P\), as shown in the diagram. Given that the forces are in equilibrium, find the values of \(F\) and \(\alpha\).

5
\includegraphics[max width=\textwidth, alt={}, center]{a92f97e2-343f-4cac-ae38-f18a4ad49055-3_574_1205_260_470} The diagram shows a velocity-time graph which models the motion of a cyclist. The graph consists of five straight line segments. The cyclist accelerates from rest to a speed of \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) over a period of 10 s , and then travels at this speed for a further 20 s . The cyclist then descends a hill, accelerating to speed \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) over a period of 10 s . This speed is maintained for a further 30 s . The cyclist then decelerates to rest over a period of 20 s .

1. Find the acceleration of the cyclist during the first 10 seconds.
2. Show that the total distance travelled by the cyclist in the 90 seconds of motion may be expressed as \(( 45 V + 150 ) \mathrm { m }\). Hence find \(V\), given that the total distance travelled by the cyclist is 465 m .
3. The combined mass of the cyclist and the bicycle is 80 kg . The cyclist experiences a constant resistance to motion of 20 N . Use an energy method to find the vertical distance which the cyclist descends during the downhill section from \(t = 30\) to \(t = 40\), assuming that the cyclist does no work during this time.

6 A block of mass 25 kg is pulled along horizontal ground by a force of magnitude 50 N inclined at \(10 ^ { \circ }\) above the horizontal. The block starts from rest and travels a distance of 20 m . There is a constant resistance force of magnitude 30 N opposing motion.

1. Find the work done by the pulling force.
2. Use an energy method to find the speed of the block when it has moved a distance of 20 m .
3. Find the greatest power exerted by the 50 N force.
   \includegraphics[max width=\textwidth, alt={}, center]{a92f97e2-343f-4cac-ae38-f18a4ad49055-3_236_1027_2161_566} After the block has travelled the 20 m , it comes to a plane inclined at \(5 ^ { \circ }\) to the horizontal. The force of 50 N is now inclined at an angle of \(10 ^ { \circ }\) to the plane and pulls the block directly up the plane (see diagram). The resistance force remains 30 N .
4. Find the time it takes for the block to come to rest from the instant when it reaches the foot of the inclined plane.
   [0pt] [4]

7 A racing car is moving in a straight line. The acceleration \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\) at time \(t \mathrm {~s}\) after the car starts from rest is given by $$\begin{array} { l l } a = 15 t - 3 t ^ { 2 } & \text { for } 0 \leqslant t \leqslant 5
a = - \frac { 625 } { t ^ { 2 } } & \text { for } 5 < t \leqslant k \end{array}$$ where \(k\) is a constant.

1. Find the maximum acceleration of the car in the first five seconds of its motion.
2. Find the distance of the car from its starting point when \(t = 5\).
3. The car comes to rest when \(t = k\). Find the value of \(k\).