OCR M1 (Mechanics 1) 2010 June

1 A block \(B\) of mass 3 kg moves with deceleration \(1.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) in a straight line on a rough horizontal surface. The initial speed of \(B\) is \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Calculate

1. the time for which \(B\) is in motion,
2. the distance travelled by \(B\) before it comes to rest,
3. the coefficient of friction between \(B\) and the surface.

2 Two particles \(P\) and \(Q\) are moving in opposite directions in the same straight line on a smooth horizontal surface when they collide. \(P\) has mass 0.4 kg and speed \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 } . Q\) has mass 0.6 kg and speed \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Immediately after the collision, the speed of \(P\) is \(0.1 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Given that \(P\) and \(Q\) are moving in the same direction after the collision, find the speed of \(Q\).
2. Given instead that \(P\) and \(Q\) are moving in opposite directions after the collision, find the distance between them 3 s after the collision.

3
\includegraphics[max width=\textwidth, alt={}, center]{4b703cf9-b3d3-4210-b57b-89136595f8a5-02_570_495_1114_826} Three horizontal forces of magnitudes \(12 \mathrm {~N} , 5 \mathrm {~N}\), and 9 N act along bearings \(000 ^ { \circ } , 150 ^ { \circ }\) and \(270 ^ { \circ }\) respectively (see diagram).

1. Show that the component of the resultant of the three forces along bearing \(270 ^ { \circ }\) has magnitude 6.5 N .
2. Find the component of the resultant of the three forces along bearing \(000 ^ { \circ }\).
3. Hence find the magnitude and bearing of the resultant of the three forces.

4 A particle \(P\) moving in a straight line has velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at time \(t \mathrm {~s}\) after passing through a fixed point \(O\). It is given that \(v = 3.2 - 0.2 t ^ { 2 }\) for \(0 \leqslant t \leqslant 5\). Calculate

1. the value of \(t\) when \(P\) is at instantaneous rest,
2. the acceleration of \(P\) when it is at instantaneous rest,
3. the greatest distance of \(P\) from \(O\).

5
\includegraphics[max width=\textwidth, alt={}, center]{4b703cf9-b3d3-4210-b57b-89136595f8a5-03_508_1397_255_374} The diagram shows the ( \(t , v\) ) graph for a lorry delivering waste to a recycling centre. The graph consists of six straight line segments. The lorry reverses in a straight line from a stationary position on a weighbridge before coming to rest. It deposits its waste and then moves forwards in a straight line accelerating to a maximum speed of \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). It maintains this speed for 4 s and then decelerates, coming to rest at the weighbridge.

1. Calculate the distance from the weighbridge to the point where the lorry deposits the waste.
2. Calculate the time which elapses between the lorry leaving the weighbridge and returning to it.
3. Given that the acceleration of the lorry when it is moving forwards is \(0.4 \mathrm {~m} \mathrm {~s} ^ { - 2 }\), calculate its final deceleration.

6 A block \(B\) of mass 0.85 kg lies on a smooth slope inclined at \(30 ^ { \circ }\) to the horizontal. \(B\) is attached to one end of a light inextensible string which is parallel to the slope. At the top of the slope, the string passes over a smooth pulley. The other end of the string hangs vertically and is attached to a particle \(P\) of mass 0.55 kg . The string is taut at the instant when \(P\) is projected vertically downwards.

1. Calculate
   (a) the acceleration of \(B\) and the tension in the string,
   (b) the magnitude of the force exerted by the string on the pulley. The initial speed of \(P\) is \(1.3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and after moving \(1.5 \mathrm {~m} P\) reaches the ground, where it remains at rest. \(B\) continues to move up the slope and does not reach the pulley.
2. Calculate the total distance \(B\) moves up the slope before coming instantaneously to rest.

7 \begin{figure}[h] \end{figure} A rectangular block \(B\) of weight 12 N lies in limiting equilibrium on a horizontal surface. A horizontal force of 4 N and a coplanar force of 5 N inclined at \(60 ^ { \circ }\) to the vertical act on \(B\) (see Fig. 1).

1. Find the coefficient of friction between \(B\) and the surface. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{4b703cf9-b3d3-4210-b57b-89136595f8a5-04_307_751_1000_696} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure} \(B\) is now cut horizontally into two smaller blocks. The upper block has weight 9 N and the lower block has weight 3 N . The 5 N force now acts on the upper block and the 4 N force now acts on the lower block (see Fig. 2). The coefficient of friction between the two blocks is \(\mu\).
2. Given that the upper block is in limiting equilibrium, find \(\mu\).
3. Given instead that \(\mu = 0.1\), find the accelerations of the two blocks.

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