OCR MEI M1 (Mechanics 1) 2005 June

1 A particle travels along a straight line. Its acceleration during the time interval \(0 \leqslant t \leqslant 8\) is given by the acceleration-time graph in Fig. 1. \begin{figure}[h] \end{figure}

1. Write down the acceleration of the particle when \(t = 4\). Given that the particle starts from rest, find its speed when \(t = 4\).
2. Write down an expression in terms of \(t\) for the acceleration, \(a \mathrm {~ms} ^ { - 2 }\), of the particle in the time interval \(0 \leqslant t \leqslant 4\).
3. Without calculation, state the time at which the speed of the particle is greatest. Give a reason for your answer.
4. Calculate the change in speed of the particle from \(t = 5\) to \(t = 8\), indicating whether this is an increase or a decrease.

2 A particle moves along the \(x\)-axis with velocity, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), at time \(t\) given by $$v = 24 t - 6 t ^ { 2 }$$ The positive direction is in the sense of \(x\) increasing.

1. Find an expression for the acceleration of the particle at time \(t\).
2. Find the times, \(t _ { 1 }\) and \(t _ { 2 }\), at which the particle has zero speed.
3. Find the distance travelled between the times \(t _ { 1 }\) and \(t _ { 2 }\).

3 A particle rests on a smooth, horizontal plane. Horizontal unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) lie in this plane. The particle is in equilibrium under the action of the three forces \(( - 3 \mathbf { i } + 4 \mathbf { j } ) \mathrm { N }\) and \(( 21 \mathbf { i } - 7 \mathbf { j } ) \mathrm { N }\) and \(\mathbf { R N }\).

1. Write down an expression for \(\mathbf { R }\) in terms of \(\mathbf { i }\) and \(\mathbf { j }\).
2. Find the magnitude of \(\mathbf { R }\) and the angle between \(\mathbf { R }\) and the \(\mathbf { i }\) direction.

4 A block of mass 4 kg is in equilibrium on a rough plane inclined at \(60 ^ { \circ }\) to the horizontal, as shown in Fig. 4. A frictional force of 10 N acts up the plane and a vertical string AB attached to the block is in tension. \begin{figure}[h] \end{figure}

1. Draw a diagram showing the four forces acting on the block.
2. By considering the components of the forces parallel to the slope, calculate the tension in the string.
3. Calculate the normal reaction of the plane on the block.

5 The position vector of a particle at time \(t\) is given by $$\mathbf { r } = \frac { 1 } { 2 } t \mathbf { i } + \left( t ^ { 2 } - 1 \right) \mathbf { j } ,$$ referred to an origin \(\mathbf { O }\) where \(\mathbf { i }\) and \(\mathbf { j }\) are the standard unit vectors in the directions of the cartesian axes \(\mathrm { O } x\) and Oy respectively.

1. Write down the value of \(t\) for which the \(x\)-coordinate of the position of the particle is 2 . Find the \(y\)-coordinate at this time.
2. Show that the cartesian equation of the path of the particle is \(y = 4 x ^ { 2 } - 1\).
3. Find the coordinates of the point where the particle is moving at \(45 ^ { \circ }\) to both \(\mathrm { O } x\) and \(\mathrm { O } y\). Section B (36 marks)