SPS SPS SM Mechanics (SPS SM Mechanics) 2023 January

1. 10 seconds after passing a warning signal, a train is travelling at \(18 m s ^ { - 1 }\) and has gone 215 m beyond the signal. Find the acceleration (assumed to be constant) of the train during the 10 seconds and its velocity as it passed the signal.

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2. A particle of mass \(m\) is placed on a rough inclined plane.
The plane makes an angle \(\theta\) with the horizontal.
The coefficient of friction between the particle and the plane is \(\mu\) where \(\mu < \tan \theta\). The particle is released from rest and accelerates down the plane.

1. Draw a fully labelled diagram to show the forces acting on the particle.
2. Find an expression in terms of \(g , \theta\) and \(\mu\) for the acceleration of the particle.
3. Explain what would happen to the particle if \(\mu > \tan \theta\). \section*{BLANK PAGE FOR WORKING}

1. A parcel \(P\) of weight 50 N is being held in equilibrium by two light, inextensible strings \(A P\) and \(B P\). The string \(A P\) is attached to a wall at \(A\), and string \(B P\) passes over a smooth pulley which is at the same height as \(A\), as shown in the diagram.

When the tension in \(B P\) is 40 N , the strings are at right angles to each other.
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1. Find the tension in string \(A P\).
2. Explain why the parcel can never be in equilibrium, with both strings horizontal. \section*{BLANK PAGE FOR WORKING}

4. \begin{figure}[h] \end{figure} A small ball, \(P\), of mass 0.8 kg , is held at rest on a smooth horizontal table and is attached to one end of a thin rope. The rope passes over a pulley that is fixed at the edge of the table.
The other end of the rope is attached to another small ball, \(Q\), of mass 0.6 kg , that hangs freely below the pulley. Ball \(P\) is released from rest, with the rope taut, with \(P\) at a distance of 1.5 m from the pulley and with \(Q\) at a height of 0.4 m above the horizontal floor, as shown in Figure 1. Ball \(Q\) descends, hits the floor and does not rebound.
The balls are modelled as particles, the rope as a light and inextensible string and the pulley as small and smooth. Using this model,

1. show that the acceleration of \(Q\), as it falls, is \(4.2 \mathrm {~ms} ^ { - 2 }\)
2. find the time taken by \(P\) to hit the pulley from the instant when \(P\) is released.
3. State one limitation of the model that will affect the accuracy of your answer to part (a). \section*{BLANK PAGE FOR WORKING}

1. At time \(t\) seconds, where \(0 \leq t \leq T\), a particle, \(P\), moves so that its velocity \(v m s ^ { - 1 }\) is given by

$$v = 7.2 t - 0.45 t ^ { 2 }$$ When \(t = 0\) the particle is sitting stationary at a displacement of \(\mathrm { d } m\) from a point O .
The particle's acceleration is zero when \(t = T\).

1. Find the value of \(T\). For \(t \geq T\), the particle moves with a velocity \(v = 48 - 2.4 t m s ^ { - 1 }\).
2. Find the time when \(P\) is at its maximum displacement from O . The particle passes through the point O when \(t = 38\).
3. Find \(d\). \section*{BLANK PAGE FOR WORKING}

6. \begin{figure}[h] \end{figure} Two blocks, \(A\) and \(B\), of masses \(2 m\) and \(3 m\) respectively, are attached to the ends of a light string. Initially \(A\) is held at rest on a fixed rough plane.
The plane is inclined at angle \(\alpha\) to the horizontal ground, where \(\tan \alpha = \frac { 5 } { 12 }\)
The string passes over a small smooth pulley, \(P\), fixed at the top of the plane.
The part of the string from \(A\) to \(P\) is parallel to a line of greatest slope of the plane.
Block \(B\) hangs freely below \(P\), as shown in Figure 1.
The coefficient of friction between \(A\) and the plane is \(\frac { 2 } { 3 }\)
The blocks are released from rest with the string taut and \(A\) moves up the plane.
The tension in the string immediately after the blocks are released is \(T\).
The blocks are modelled as particles and the string is modelled as being inextensible.

1. Show that \(T = \frac { 12 m g } { 5 }\) After \(B\) reaches the ground, \(A\) continues to move up the plane until it comes to rest before reaching \(P\).
2. Determine whether \(A\) will remain at rest, carefully justifying your answer.
3. Suggest two refinements to the model that would make it more realistic. \section*{BLANK PAGE FOR WORKING}