CAIE M1 (Mechanics 1) 2013 November

1
\includegraphics[max width=\textwidth, alt={}, center]{3e58aa5a-3789-4aaf-8656-b5b98cd7f693-2_291_591_255_776} A particle \(P\) of mass 0.3 kg is attached to one end of a light inextensible string. The other end of the string is attached to a fixed point \(X\). A horizontal force of magnitude \(F \mathrm {~N}\) is applied to the particle, which is in equilibrium when the string is at an angle \(\alpha\) to the vertical, where \(\tan \alpha = \frac { 8 } { 15 }\) (see diagram). Find the tension in the string and the value of \(F\).

2
\includegraphics[max width=\textwidth, alt={}, center]{3e58aa5a-3789-4aaf-8656-b5b98cd7f693-2_385_389_918_879} A block \(B\) lies on a rough horizontal plane. Horizontal forces of magnitudes 30 N and 40 N , making angles of \(\alpha\) and \(\beta\) respectively with the \(x\)-direction, act on \(B\) as shown in the diagram, and \(B\) is moving in the \(x\)-direction with constant speed. It is given that \(\cos \alpha = 0.6\) and \(\cos \beta = 0.8\).

1. Find the total work done by the forces shown in the diagram when \(B\) has moved a distance of 20 m .
2. Given that the coefficient of friction between the block and the plane is \(\frac { 5 } { 8 }\), find the weight of the block.

3 A cyclist exerts a constant driving force of magnitude \(F \mathrm {~N}\) while moving up a straight hill inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 36 } { 325 }\). A constant resistance to motion of 32 N acts on the cyclist. The total weight of the cyclist and his bicycle is 780 N . The cyclist's acceleration is \(- 0.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

1. Find the value of \(F\). The cyclist's speed is \(7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at the bottom of the hill.
2. Find how far up the hill the cyclist travels before coming to rest.

4 Particles \(P\) and \(Q\) are moving in a straight line on a rough horizontal plane. The frictional forces are the only horizontal forces acting on the particles.

1. Find the deceleration of each of the particles given that the coefficient of friction between \(P\) and the plane is 0.2 , and between \(Q\) and the plane is 0.25 . At a certain instant, \(P\) passes through the point \(A\) and \(Q\) passes through the point \(B\). The distance \(A B\) is 5 m . The velocities of \(P\) and \(Q\) at \(A\) and \(B\) are \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), respectively, both in the direction \(A B\).
2. Find the speeds of \(P\) and \(Q\) immediately before they collide.

5 A lorry of mass 15000 kg climbs from the bottom to the top of a straight hill, of length 1440 m , at a constant speed of \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The top of the hill is 16 m above the level of the bottom of the hill. The resistance to motion is constant and equal to 1800 N .

1. Find the work done by the driving force. On reaching the top of the hill the lorry continues on a straight horizontal road and passes through a point \(P\) with speed \(24 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The resistance to motion is constant and is now equal to 1600 N . The work done by the lorry's engine from the top of the hill to the point \(P\) is 5030 kJ .
2. Find the distance from the top of the hill to the point \(P\).

6
\includegraphics[max width=\textwidth, alt={}, center]{3e58aa5a-3789-4aaf-8656-b5b98cd7f693-3_518_515_1436_815} Particles \(A\) and \(B\), of masses 0.3 kg and 0.7 kg respectively, are attached to the ends of a light inextensible string. The string passes over a fixed smooth pulley. \(A\) is held at rest and \(B\) hangs freely, with both straight parts of the string vertical and both particles at a height of 0.52 m above the floor (see diagram). \(A\) is released and both particles start to move.

1. Find the tension in the string. When both particles are moving with speed \(1.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) the string breaks.
2. Find the time taken, from the instant that the string breaks, for \(A\) to reach the floor.
   \(7 \quad\) A particle \(P\) starts from rest at a point \(O\) and moves in a straight line. \(P\) has acceleration \(0.6 t \mathrm {~m} \mathrm {~s} ^ { - 2 }\) at time \(t\) seconds after leaving \(O\), until \(t = 10\).
3. Find the velocity and displacement from \(O\) of \(P\) when \(t = 10\). After \(t = 10 , P\) has acceleration \(- 0.4 t \mathrm {~m} \mathrm {~s} ^ { - 2 }\) until it comes to rest at a point \(A\).
4. Find the distance \(O A\).