CAIE M1 (Mechanics 1) 2011 June

1 A load is pulled along horizontal ground for a distance of 76 m , using a rope. The rope is inclined at \(5 ^ { \circ }\) above the horizontal and the tension in the rope is 65 N .

1. Find the work done by the tension. At an instant during the motion the velocity of the load is \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
2. Find the rate of working of the tension at this instant.

2 An object of mass 8 kg slides down a line of greatest slope of an inclined plane. Its initial speed at the top of the plane is \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and its speed at the bottom of the plane is \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The work done against the resistance to motion of the object is 120 J . Find the height of the top of the plane above the level of the bottom.

3
\includegraphics[max width=\textwidth, alt={}, center]{d3bb6702-231d-42a0-830e-9f844dca78d7-2_748_1410_979_370} The velocity-time graph shown models the motion of a parachutist falling vertically. There are four stages in the motion:

• falling freely with the parachute closed,
• decelerating at a constant rate with the parachute open,
• falling with constant speed with the parachute open,
• coming to rest instantaneously on hitting the ground.
  1. Show that the total distance fallen is 1048 m .

The weight of the parachutist is 850 N .

Find the upward force on the parachutist due to the parachute, during the second stage.

4
\includegraphics[max width=\textwidth, alt={}, center]{d3bb6702-231d-42a0-830e-9f844dca78d7-3_511_901_255_621} The three coplanar forces shown in the diagram act at a point \(P\) and are in equilibrium.

1. Find the values of \(F\) and \(\theta\).
2. State the magnitude and direction of the resultant force at \(P\) when the force of magnitude 12 N is removed.

5 Two particles \(P\) and \(Q\) are projected vertically upwards from horizontal ground at the same instant. The speeds of projection of \(P\) and \(Q\) are \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively and the heights of \(P\) and \(Q\) above the ground, \(t\) seconds after projection, are \(h _ { P } \mathrm {~m}\) and \(h _ { Q } \mathrm {~m}\) respectively. Each particle comes to rest on returning to the ground.

1. Find the set of values of \(t\) for which the particles are travelling in opposite directions.
2. At a certain instant, \(P\) and \(Q\) are above the ground and \(3 h _ { P } = 8 h _ { Q }\). Find the velocities of \(P\) and \(Q\) at this instant.

6
\includegraphics[max width=\textwidth, alt={}, center]{d3bb6702-231d-42a0-830e-9f844dca78d7-3_387_1095_1724_525} A small smooth ring \(R\), of mass 0.6 kg , is threaded on a light inextensible string of length 100 cm . One end of the string is attached to a fixed point \(A\). A small bead \(B\) of mass 0.4 kg is attached to the other end of the string, and is threaded on a fixed rough horizontal rod which passes through \(A\). The system is in equilibrium with \(B\) at a distance of 80 cm from \(A\) (see diagram).

1. Find the tension in the string.
2. Find the frictional and normal components of the contact force acting on \(B\).
3. Given that the equilibrium is limiting, find the coefficient of friction between the bead and the rod.

7 A walker travels along a straight road passing through the points \(A\) and \(B\) on the road with speeds \(0.9 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(1.3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively. The walker's acceleration between \(A\) and \(B\) is constant and equal to \(0.004 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

1. Find the time taken by the walker to travel from \(A\) to \(B\), and find the distance \(A B\). A cyclist leaves \(A\) at the same instant as the walker. She starts from rest and travels along the straight road, passing through \(B\) at the same instant as the walker. At time \(t \mathrm {~s}\) after leaving \(A\) the cyclist's speed is \(k t ^ { 3 } \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where \(k\) is a constant.
2. Show that when \(t = 64.05\) the speed of the walker and the speed of the cyclist are the same, correct to 3 significant figures.
3. Find the cyclist's acceleration at the instant she passes through \(B\).