OCR MEI M1 (Mechanics 1) 2016 June

1 Fig. 1 shows a block of mass \(M \mathrm {~kg}\) being pushed over level ground by means of a light rod. The force, \(T \mathrm {~N}\), this exerts on the block is along the line of the rod. The ground is rough.
The rod makes an angle \(\alpha\) with the horizontal. \begin{figure}[h] \end{figure}

1. Draw a diagram showing all the forces acting on the block.
2. You are given that \(M = 5 , \alpha = 60 ^ { \circ } , T = 40\) and the acceleration of the block is \(1.5 \mathrm {~ms} ^ { - 2 }\). Find the frictional force.

2 \begin{figure}[h] \end{figure} A particle moves on the straight line shown in Fig. 2. The positive direction is indicated on the diagram. The time, \(t\), is measured in seconds. The particle has constant acceleration, \(a \mathrm {~ms} ^ { - 2 }\). Initially it is at the point O and has velocity \(u \mathrm {~ms} ^ { - 1 }\).
When \(t = 2\), the particle is at A where OA is 12 m . The particle is also at A when \(t = 6\).

1. Write down two equations in \(u\) and \(a\) and solve them.
2. The particle changes direction when it is at B . Find the distance AB .

3 Fig. 3.1 shows a block of mass 8 kg on a smooth horizontal table.
This block is connected by a light string passing over a smooth pulley to a block of mass 4 kg which hangs freely. The part of the string between the 8 kg block and the pulley is parallel to the table. The system has acceleration \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\). \begin{figure}[h] \end{figure}

1. Write down two equations of motion, one for each block.
2. Find the value of \(a\). The table is now tilted at an angle of \(\theta\) to the horizontal as shown in Fig. 3.2. The system is set up as before; the 4 kg block still hangs freely. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{4c8c96cf-5184-46e4-9c45-a8a80d0a6ff8-3_410_727_1324_669} \captionsetup{labelformat=empty} \caption{Fig. 3.2}
   \end{figure}
3. The system is now in equilibrium. Find the value of \(\theta\).

4 A particle is initially at the origin, moving with velocity \(\mathbf { u }\). Its acceleration \(\mathbf { a }\) is constant. At time \(t\) its displacement from the origin is \(\mathbf { r } = \binom { x } { y }\), where \(\binom { x } { y } = \binom { 2 } { 6 } t - \binom { 0 } { 4 } t ^ { 2 }\).

1. Write down \(\mathbf { u }\) and \(\mathbf { a }\) as column vectors.
2. Find the speed of the particle when \(t = 2\).
3. Show that the equation of the path of the particle is \(y = 3 x - x ^ { 2 }\).

5 Mr McGregor is a keen vegetable gardener. A pigeon that eats his vegetables is his great enemy.
One day he sees the pigeon sitting on a small branch of a tree. He takes a stone from the ground and throws it. The trajectory of the stone is in a vertical plane that contains the pigeon. The same vertical plane intersects the window of his house. The situation is illustrated in Fig. 5. \begin{figure}[h] \end{figure}

• The stone is thrown from point O on level ground. Its initial velocity is \(15 \mathrm {~ms} ^ { - 1 }\) in the horizontal direction and \(8 \mathrm {~ms} ^ { - 1 }\) in the vertical direction.
• The pigeon is at point P which is 4 m above the ground.
• The house is 22.5 m from O .
• The bottom of the window is 0.8 m above the ground and the window is 1.2 m high.

Show that the stone does not reach the height of the pigeon. Determine whether the stone hits the window.

6 In this question you should take \(\boldsymbol { g \) to be \(\mathbf { 1 0 } \mathrm { ms } ^ { \boldsymbol { - } \mathbf { 2 } }\).} Piran finds a disused mineshaft on his land and wants to know its depth, \(d\) metres.
Local records state that the mineshaft is between 150 and 200 metres deep.
He drops a small stone down the mineshaft and records the time, \(T\) seconds, until he hears it hit the bottom. It takes 8.0 seconds. Piran tries three models, \(\mathrm { A } , \mathrm { B }\) and C .
In model A, Piran uses the formula \(d = 5 T ^ { 2 }\) to estimate the depth.

1. Find the depth that model A gives and comment on whether it is consistent with the local records. Explain how the formula in model A is obtained. In model B, Piran uses the speed-time graph in Fig. 6. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{4c8c96cf-5184-46e4-9c45-a8a80d0a6ff8-5_762_1176_1087_424} \captionsetup{labelformat=empty} \caption{Fig. 6}
   \end{figure}
2. Calculate the depth of the mineshaft according to model B. Comment on whether this depth is consistent with the local records.
3. Describe briefly one respect in which model B is the same as model A and one respect in which it is different. Piran then tries model C in which the speed, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), is given by $$\begin{aligned} & v = 10 t - t ^ { 2 } \text { for } 0 \leqslant t \leqslant 5
   & v = 25 \text { for } 5 < t \leqslant 8 \end{aligned}$$
4. Calculate the depth of the mineshaft according to model C. Comment on whether this depth is consistent with the local records.
5. Describe briefly one respect in which model C is similar to model B and one respect in which it is different.