OCR MEI M1 (Mechanics 1) 2010 January

1 A ring is moving up and down a vertical pole. The displacement, \(s \mathrm {~m}\), of the ring above a mark on the pole is modelled by the displacement-time graph shown in Fig. 1. The three sections of the graph are straight lines. \begin{figure}[h] \end{figure}

1. Calculate the velocity of the ring in the interval \(0 < t < 2\) and in the interval \(2 < t < 3.5\).
2. Sketch a velocity-time graph for the motion of the ring during the 4 seconds.
3. State the direction of motion of the ring when
   (A) \(t = 1\),
   (B) \(t = 2.75\),
   (C) \(t = 3.25\).

2 A particle of mass 5 kg has constant acceleration. Initially, the particle is at \(\binom { - 1 } { 2 } \mathrm {~m}\) with velocity \(\binom { 2 } { - 3 } \mathrm {~m} \mathrm {~s} ^ { - 1 }\); after 4 seconds the particle has velocity \(\binom { 12 } { 9 } \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Calculate the acceleration of the particle.
2. Calculate the position of the particle at the end of the 4 seconds.
3. Calculate the force acting on the particle.

3 In this question, \(\mathbf { i }\) is a horizontal unit vector and \(\mathbf { j }\) is a unit vector pointing vertically upwards.
A force \(\mathbf { F }\) is \(- \mathbf { i } + 5 \mathbf { j }\).

1. Calculate the magnitude of \(\mathbf { F }\). Calculate also the angle between \(\mathbf { F }\) and the upward vertical. Force \(\mathbf { G }\) is \(2 a \mathbf { i } + a \mathbf { j }\) and force \(\mathbf { H }\) is \(- 2 \mathbf { i } + 3 b \mathbf { j }\), where \(a\) and \(b\) are constants. The force \(\mathbf { H }\) is the resultant of forces \(4 \mathbf { F }\) and \(\mathbf { G }\).
2. Find \(\mathbf { G }\) and \(\mathbf { H }\).

4 A box of mass 2.5 kg is on a smooth horizontal table, as shown in Fig. 4. A light string AB is attached to the table at A and the box at B . AB is at an angle of \(50 ^ { \circ }\) to the vertical. Another light string is attached to the box at C ; this string is inclined at \(15 ^ { \circ }\) above the horizontal and the tension in it is 20 N . The box is in equilibrium. \begin{figure}[h] \end{figure}

1. Calculate the horizontal component of the force exerted on the box by the string at C .
2. Calculate the tension in the string AB .
3. Calculate the normal reaction of the table on the box. The string at C is replaced by one inclined at \(15 ^ { \circ }\) below the horizontal with the same tension of 20 N .
4. Explain why this has no effect on the tension in string AB .