OCR MEI M1 (Mechanics 1) 2005 January

1 The position vector, \(\mathbf { r }\), of a particle of mass 4 kg at time \(t\) is given by $$\mathbf { r } = t ^ { 2 } \mathbf { i } + \left( 5 t - 2 t ^ { 2 } \right) \mathbf { j } ,$$ where \(\mathbf { i }\) and \(\mathbf { j }\) are the standard unit vectors, lengths are in metres and time is in seconds.

1. Find an expression for the acceleration of the particle. The particle is subject to a force \(\mathbf { F }\) and a force \(12 \mathbf { j } \mathbf { N }\).
2. Find \(\mathbf { F }\).

2 Particles of mass 2 kg and 4 kg are attached to the ends \(X\) and \(Y\) of a light, inextensible string. The string passes round fixed, smooth pulleys at \(\mathrm { P } , \mathrm { Q }\) and R , as shown in Fig. 2. The system is released from rest with the string taut. \begin{figure}[h] \end{figure}

1. State what information in the question tells you that
   (A) the tension is the same throughout the string,
   (B) the magnitudes of the accelerations of the particles at X and Y are the same. The tension in the string is \(T \mathrm {~N}\) and the magnitude of the acceleration of the particles is \(a \mathrm {~ms} ^ { - 2 }\).
2. Draw a diagram showing the forces acting at X and a diagram showing the forces acting at Y .
3. Write down equations of motion for the particles at X and at Y . Hence calculate the values of \(T\) and \(a\).

3 A particle is in equilibrium when acted on by the forces \(\left( \begin{array} { r } x
- 7
z \end{array} \right) , \left( \begin{array} { r } 4
y
- 5 \end{array} \right)\) and \(\left( \begin{array} { r } 5
4
- 7 \end{array} \right)\), where the units are newtons.

1. Find the values of \(x , y\) and \(z\).
2. Calculate the magnitude of \(\left( \begin{array} { r } 5
   4
   - 7 \end{array} \right)\).

4 A particle is projected vertically upwards from a point O at \(21 \mathrm {~ms} ^ { - 1 }\).

1. Calculate the greatest height reached by the particle. When this particle is at its highest point, a second particle is projected vertically upwards from \(O\) at \(15 \mathrm {~ms} ^ { - 1 }\).
2. Show that the particles collide 1.5 seconds later and determine the height above O at which the collision takes place.

5 A small box B of weight 400 N is held in equilibrium by two light strings AB and BC . The string BC is fixed at C . The end A of string AB is fixed so that AB is at an angle \(\alpha\) to the vertical where \(\alpha < 60 ^ { \circ }\). String BC is at \(60 ^ { \circ }\) to the vertical. This information is shown in Fig. 5. \begin{figure}[h] \end{figure}

1. Draw a labelled diagram showing all the forces acting on the box.
2. In one situation string AB is fixed so that \(\alpha = 30 ^ { \circ }\). By drawing a triangle of forces, or otherwise, calculate the tension in the string BC and the tension in the string AB .
3. Show carefully, but briefly, that the box cannot be in equilibrium if \(\alpha = 60 ^ { \circ }\) and BC remains at \(60 ^ { \circ }\) to the vertical.

6 In this question take \(g\) as \(10 \mathrm {~m \mathrm {~s} ^ { - 2 }\).} A small ball is released from rest. It falls for 2 seconds and is then brought to rest over the next 5 seconds. This motion is modelled in the speed-time graph Fig. 6. \begin{figure}[h] \end{figure} For this model,

1. calculate the distance fallen from \(t = 0\) to \(t = 7\),
2. find the acceleration of the ball from \(t = 2\) to \(t = 6\), specifying the direction,
3. obtain an expression in terms of \(t\) for the downward speed of the ball from \(t = 2\) to \(t = 6\),
4. state the assumption that has been made about the resistance to motion from \(t = 0\) to \(t = 2\). The part of the motion from \(t = 2\) to \(t = 7\) is now modelled by \(v = - \frac { 3 } { 2 } t ^ { 2 } + \frac { 19 } { 2 } t + 7\).
5. Verify that \(v\) agrees with the values given in Fig. 6 at \(t = 2 , t = 6\) and \(t = 7\).
6. Calculate the distance fallen from \(t = 2\) to \(t = 7\) according to this model.

7 The trajectory ABCD of a small stone moving with negligible air resistance is shown in Fig. 7. AD is horizontal and BC is parallel to AD . The stone is projected from A with speed \(40 \mathrm {~ms} ^ { - 1 }\) at \(50 ^ { \circ }\) to the horizontal. \begin{figure}[h] \end{figure}

1. Write down an expression for the horizontal displacement from A of the stone \(t\) seconds after projection. Write down also an expression for the vertical displacement at time \(t\).
2. Show that the stone takes 6.253 seconds (to three decimal places) to travel from A to D . Calculate the range of the stone. You are given that \(X = 30\).
3. Calculate the time it takes the stone to reach B . Hence determine the time for it to travel from A to C.
4. Calculate the direction of the motion of the stone at \(\mathbf { C }\). Section B (36 marks)