CAIE M1 (Mechanics 1) 2016 June

1 A lift moves upwards from rest and accelerates at \(0.9 \mathrm {~ms} ^ { - 2 }\) for 3 s . The lift then travels for 6 s at constant speed and finally slows down, with a constant deceleration, stopping in a further 4 s .

1. Sketch a velocity-time graph for the motion.
2. Find the total distance travelled by the lift.

2 A box of mass 25 kg is pulled, at a constant speed, a distance of 36 m up a rough plane inclined at an angle of \(20 ^ { \circ }\) to the horizontal. The box moves up a line of greatest slope against a constant frictional force of 40 N . The force pulling the box is parallel to the line of greatest slope. Find

1. the work done against friction,
2. the change in gravitational potential energy of the box,
3. the work done by the pulling force.

3 A car of mass 1000 kg is moving along a straight horizontal road against resistances of total magnitude 300 N .

1. Find, in kW , the rate at which the engine of the car is working when the car has a constant speed of \(40 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
2. Find the acceleration of the car when its speed is \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the engine is working at \(90 \%\) of the power found in part (i).

4
\includegraphics[max width=\textwidth, alt={}, center]{099c81e0-a95a-4f98-801c-32d905ef7c7d-2_446_752_1521_699} Coplanar forces of magnitudes \(50 \mathrm {~N} , 48 \mathrm {~N} , 14 \mathrm {~N}\) and \(P \mathrm {~N}\) act at a point in the directions shown in the diagram. The system is in equilibrium. Given that \(\tan \alpha = \frac { 7 } { 24 }\), find the values of \(P\) and \(\theta\).

5
\includegraphics[max width=\textwidth, alt={}, center]{099c81e0-a95a-4f98-801c-32d905ef7c7d-3_432_710_258_721} Two particles of masses 5 kg and 10 kg are connected by a light inextensible string that passes over a fixed smooth pulley. The 5 kg particle is on a rough fixed slope which is at an angle of \(\alpha\) to the horizontal, where \(\tan \alpha = \frac { 3 } { 4 }\). The 10 kg particle hangs below the pulley (see diagram). The coefficient of friction between the slope and the 5 kg particle is \(\frac { 1 } { 2 }\). The particles are released from rest. Find the acceleration of the particles and the tension in the string.

6 A particle \(P\) moves in a straight line. It starts at a point \(O\) on the line and at time \(t\) s after leaving \(O\) it has a velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where \(v = 6 t ^ { 2 } - 30 t + 24\).

1. Find the set of values of \(t\) for which the acceleration of the particle is negative.
2. Find the distance between the two positions at which \(P\) is at instantaneous rest.
3. Find the two positive values of \(t\) at which \(P\) passes through \(O\).

7 A particle of mass 30 kg is on a plane inclined at an angle of \(20 ^ { \circ }\) to the horizontal. Starting from rest, the particle is pulled up the plane by a force of magnitude 200 N acting parallel to a line of greatest slope.

1. Given that the plane is smooth, find
   (a) the acceleration of the particle,
   (b) the change in kinetic energy after the particle has moved 12 m up the plane.
2. It is given instead that the plane is rough and the coefficient of friction between the particle and the plane is 0.12 .
   (a) Find the acceleration of the particle.
   (b) The direction of the force of magnitude 200 N is changed, and the force now acts at an angle of \(10 ^ { \circ }\) above the line of greatest slope. Find the acceleration of the particle.