OCR MEI M1 (Mechanics 1) 2011 June

1 A pellet is fired vertically upwards at a speed of \(11 \mathrm {~ms} ^ { - 1 }\). Assuming that air resistance may be neglected, calculate the speed at which the pellet hits a ceiling 2.4 m above its point of projection.

2 A particle travels with constant acceleration along a straight line. A and B are points on this line 8 m apart. The motion of the particle is as follows.

• Initially it is at A.
• After 32 s it is at B .
• When it is at B its speed is \(2.25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and it is moving away from A .

In either order, calculate the acceleration and the initial velocity of the particle, making the directions clear.

3 Force \(\mathbf { F }\) is \(\left( \begin{array} { r } - 2
3
- 4 \end{array} \right) \mathrm { N }\), force \(\mathbf { G }\) is \(\left( \begin{array} { r } - 6
y
z \end{array} \right) \mathrm { N }\) and force \(\mathbf { H }\) is \(\left( \begin{array} { r } 3
- 5
- 1 \end{array} \right) \mathrm { N }\).

1. Given that \(\mathbf { F }\) and \(\mathbf { G }\) act in parallel lines, find \(y\) and \(z\). Forces \(\mathbf { F }\) and \(\mathbf { H }\) are the only forces acting on an object of mass 5 kg .
2. Calculate the acceleration of the object. Calculate also the magnitude of this acceleration.

4 Fig. 4 shows a block of mass 15 kg on a smooth plane inclined at \(20 ^ { \circ }\) to the horizontal. The block is held in equilibrium by a horizontal force of magnitude \(P \mathrm {~N}\). \begin{figure}[h] \end{figure}

1. Show all the forces acting on the block.
2. Calculate \(P\).

5 A small object is projected over horizontal ground from a point O at ground level and makes a loud noise on landing. It has an initial speed of \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at \(35 ^ { \circ }\) to the horizontal. Assuming that air resistance on the object may be neglected and that the speed of sound in air is \(343 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), calculate how long after projection the noise is heard at O .

6 In this question, \(\mathbf { i }\) and \(\mathbf { j }\) are unit vectors east and north respectively. Position vectors are with respect to an origin O . Time \(t\) is in seconds. A skater has a constant acceleration of \(- 2 \mathbf { j } \mathrm {~m} \mathrm {~s} ^ { - 2 }\). At \(t = 0\), his velocity is \(4 \mathbf { i } \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and his position vector is \(3 \mathbf { j } \mathrm {~m}\).

1. Find expressions in terms of \(t\) for the velocity and the position vector of the skater at time \(t\).
2. Calculate as a bearing the direction of motion of the skater when \(t = 2.5\).

7 A ring is moving on a straight wire. Its velocity is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at time \(t\) seconds after passing a point Q . Model A for the motion of the ring gives the velocity-time graph for \(0 \leqslant t \leqslant 6\) shown in Fig. 7 . \begin{figure}[h] \end{figure} Use model A to calculate the following.

1. The acceleration of the ring when \(t = 0.5\).
2. The displacement of the ring from Q when
   (A) \(t = 2\),
   (B) \(t = 6\). In an alternative model B , the velocity of the ring is given by \(v = 2 t ^ { 2 } - 14 t + 20\) for \(0 \leqslant t \leqslant 6\).
3. Calculate the acceleration of the ring at \(t = 0.5\) as given by model B .
4. Calculate by how much the models differ in their values for the least \(v\) in the time interval \(0 \leqslant t \leqslant 6\).
5. Calculate the displacement of the ring from Q when \(t = 6\) as given by model B .

8 A trolley C of mass 8 kg with rusty axle bearings is initially at rest on a horizontal floor.
The trolley stays at rest when it is pulled by a horizontal string with tension 25 N , as shown in Fig. 8.1. \begin{figure}[h] \end{figure}

1. State the magnitude of the horizontal resistance opposing the pull. A second trolley D of mass 10 kg is connected to trolley C by means of a light, horizontal rod.
   The string now has tension 50 N , and is at an angle of \(25 ^ { \circ }\) to the horizontal, as shown in Fig. 8.2. The two trolleys stay at rest. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{2efbb554-fe60-42ce-9213-8c66bfdb1d85-5_305_1191_1050_701} \captionsetup{labelformat=empty} \caption{Fig. 8.2}
   \end{figure}
2. Calculate the magnitude of the total horizontal resistance acting on the two trolleys opposing the pull.
3. Calculate the normal reaction of the floor on trolley C . The axle bearings of the trolleys are oiled and the total horizontal resistance to the motion of the two trolleys is now 20 N . The two trolleys are still pulled by the string with tension 50 N , as shown in Fig. 8.2.
4. Calculate the acceleration of the trolleys. In a new situation, the trolleys are on a slope at \(5 ^ { \circ }\) to the horizontal and are initially travelling down the slope at \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The resistances are 15 N to the motion of D and 5 N to the motion of C . There is no string attached. The rod connecting the trolleys is parallel to the slope. This situation is shown in Fig. 8.3. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{2efbb554-fe60-42ce-9213-8c66bfdb1d85-5_355_1294_2156_429} \captionsetup{labelformat=empty} \caption{Fig. 8.3}
   \end{figure}
5. Calculate the speed of the trolleys after 2 seconds and also the force in the rod connecting the trolleys, stating whether this rod is in tension or thrust (compression).