CAIE M1 (Mechanics 1) 2010 November

1 A block of mass 400 kg rests in limiting equilibrium on horizontal ground. A force of magnitude 2000 N acts on the block at an angle of \(15 ^ { \circ }\) to the upwards vertical. Find the coefficient of friction between the block and the ground, correct to 2 significant figures.

2 A cyclist, working at a constant rate of 400 W , travels along a straight road which is inclined at \(2 ^ { \circ }\) to the horizontal. The total mass of the cyclist and his cycle is 80 kg . Ignoring any resistance to motion, find, correct to 1 decimal place, the acceleration of the cyclist when he is travelling

1. uphill at \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\),
2. downhill at \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

3
\includegraphics[max width=\textwidth, alt={}, center]{881993e1-71ea-4801-bfc8-40c17a1387a9-2_597_616_888_762} A particle \(P\) is in equilibrium on a smooth horizontal table under the action of four horizontal forces of magnitudes \(6 \mathrm {~N} , 5 \mathrm {~N} , F \mathrm {~N}\) and \(F \mathrm {~N}\) acting in the directions shown. Find the values of \(\alpha\) and \(F\).

4 A block of mass 20 kg is pulled from the bottom to the top of a slope. The slope has length 10 m and is inclined at \(4.5 ^ { \circ }\) to the horizontal. The speed of the block is \(2.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at the bottom of the slope and \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at the top of the slope.

1. Find the loss of kinetic energy and the gain in potential energy of the block.
2. Given that the work done against the resistance to motion is 50 J , find the work done by the pulling force acting on the block.
3. Given also that the pulling force is constant and acts at an angle of \(15 ^ { \circ }\) upwards from the slope, find its magnitude.

5 Particles \(P\) and \(Q\) are projected vertically upwards, from different points on horizontal ground, with velocities of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively. \(Q\) is projected 0.4 s later than \(P\). Find

1. the time for which \(P\) 's height above the ground is greater than 15 m ,
2. the velocities of \(P\) and \(Q\) at the instant when the particles are at the same height.

6
\includegraphics[max width=\textwidth, alt={}, center]{881993e1-71ea-4801-bfc8-40c17a1387a9-3_579_1518_258_315} The diagram shows the velocity-time graph for a particle \(P\) which travels on a straight line \(A B\), where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the velocity of \(P\) at time \(t \mathrm {~s}\). The graph consists of five straight line segments. The particle starts from rest when \(t = 0\) at a point \(X\) on the line between \(A\) and \(B\) and moves towards \(A\). The particle comes to rest at \(A\) when \(t = 2.5\).

1. Given that the distance \(X A\) is 4 m , find the greatest speed reached by \(P\) during this stage of the motion. In the second stage, \(P\) starts from rest at \(A\) when \(t = 2.5\) and moves towards \(B\). The distance \(A B\) is 48 m . The particle takes 12 s to travel from \(A\) to \(B\) and comes to rest at \(B\). For the first 2 s of this stage \(P\) accelerates at \(3 \mathrm {~m} \mathrm {~s} ^ { - 2 }\), reaching a velocity of \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find
2. the value of \(V\),
3. the value of \(t\) at which \(P\) starts to decelerate during this stage,
4. the deceleration of \(P\) immediately before it reaches \(B\).
   \(7 \quad\) A particle \(P\) travels in a straight line. It passes through the point \(O\) of the line with velocity \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at time \(t = 0\), where \(t\) is in seconds. \(P\) 's velocity after leaving \(O\) is given by $$\left( 0.002 t ^ { 3 } - 0.12 t ^ { 2 } + 1.8 t + 5 \right) \mathrm { m } \mathrm {~s} ^ { - 1 }$$ The velocity of \(P\) is increasing when \(0 < t < T _ { 1 }\) and when \(t > T _ { 2 }\), and the velocity of \(P\) is decreasing when \(T _ { 1 } < t < T _ { 2 }\).
5. Find the values of \(T _ { 1 }\) and \(T _ { 2 }\) and the distance \(O P\) when \(t = T _ { 2 }\).
6. Find the velocity of \(P\) when \(t = T _ { 2 }\) and sketch the velocity-time graph for the motion of \(P\).