CAIE M1 (Mechanics 1) 2016 November

1 A crane is used to raise a block of mass 50 kg vertically upwards at constant speed through a height of 3.5 m . There is a constant resistance to motion of 25 N .

1. Find the work done by the crane.
2. Given that the time taken to raise the block is 2 s , find the power of the crane.

2
\includegraphics[max width=\textwidth, alt={}, center]{94c11160-a718-4de5-867a-27c755051fa6-2_342_629_616_756} The diagram shows a small object \(P\) of mass 20 kg held in equilibrium by light ropes attached to fixed points \(A\) and \(B\). The rope \(P A\) is inclined at an angle of \(50 ^ { \circ }\) above the horizontal, the rope \(P B\) is inclined at an angle of \(10 ^ { \circ }\) below the horizontal, and both ropes are in the same vertical plane. Find the tension in the rope \(P A\) and the tension in the rope \(P B\).

3
\includegraphics[max width=\textwidth, alt={}, center]{94c11160-a718-4de5-867a-27c755051fa6-2_312_1207_1320_468} Particles \(P\) and \(Q\), of masses 7 kg and 3 kg respectively, are attached to the two ends of a light inextensible string. The string passes over two small smooth pulleys attached to the two ends of a horizontal table. The two particles hang vertically below the two pulleys. The two particles are both initially at rest, 0.5 m below the level of the table, and 0.4 m above the horizontal floor (see diagram).

1. Find the acceleration of the particles and the speed of \(P\) immediately before it reaches the floor.
2. Determine whether \(Q\) comes to instantaneous rest before it reaches the pulley directly above it.

4 A ball \(A\) is released from rest at the top of a tall tower. One second later, another ball \(B\) is projected vertically upwards from ground level near the bottom of the tower with a speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The two balls are at the same height 1.5 s after ball \(B\) is projected.

1. Show that the height of the tower is 50 m .
2. Find the length of time for which ball \(B\) has been in motion when ball \(A\) reaches the ground. Hence find the total distance travelled by ball \(B\) up to the instant when ball \(A\) reaches the ground.

5 A particle \(P\) starts from a fixed point \(O\) and moves in a straight line. At time \(t\) s after leaving \(O\), the velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) of \(P\) is given by \(v = 6 t - 0.3 t ^ { 2 }\). The particle comes to instantaneous rest at point \(X\).

1. Find the distance \(O X\). A second particle \(Q\) starts from rest from \(O\), at the same instant as \(P\), and also travels in a straight line. The acceleration \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\) of \(Q\) is given by \(a = k - 12 t\), where \(k\) is a constant. The displacement of \(Q\) from \(O\) is 400 m when \(t = 10\).
2. Find the value of \(k\).

6 A cyclist is cycling with constant power of 160 W along a horizontal straight road. There is a constant resistance to motion of 20 N . At an instant when the cyclist's speed is \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), his acceleration is \(0.15 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

1. Show that the total mass of the cyclist and bicycle is 80 kg . The cyclist comes to a hill inclined at \(2 ^ { \circ }\) to the horizontal. When the cyclist starts climbing the hill, he increases his power to a constant 300 W . The resistance to motion remains 20 N .
2. Show that the steady speed up the hill which the cyclist can maintain when working at this power is \(6.26 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), correct to 3 significant figures.
3. Find the acceleration at an instant when the cyclist is travelling at \(90 \%\) of the speed in part (ii).

7 A box of mass 50 kg is at rest on a plane inclined at \(10 ^ { \circ }\) to the horizontal.

1. Find an inequality for the coefficient of friction between the box and the plane. In fact the coefficient of friction between the box and the plane is 0.19 .
2. A girl pushes the box with a force of 50 N , acting down a line of greatest slope of the plane, for a distance of 5 m . She then stops pushing. Use an energy method to find the speed of the box when it has travelled a further 5 m . The box then comes to a plane inclined at \(20 ^ { \circ }\) below the horizontal. The box moves down a line of greatest slope of this plane. The coefficient of friction is still 0.19 and the girl is not pushing the box.
3. Find the acceleration of the box.