SPS SPS FM Mechanics (SPS FM Mechanics) 2021 June

1. A train is travelling between two stations that are 4.8 km apart on a straight horizontal track.

It accelerates uniformly from rest to a speed of \(40 \mathrm {~ms} ^ { - 1 }\) covering a distance of 400 m .
It then travels at \(40 \mathrm {~ms} ^ { - 1 }\) for \(T\) seconds and decelerates uniformly at \(0.8 \mathrm {~ms} ^ { - 2 }\) for the final part of the journey until it arrives at the next station. This is represented in the velocity-time graph below.
\includegraphics[max width=\textwidth, alt={}, center]{6c69f370-0d2d-41ec-8761-0707a6ada43d-02_595_1394_497_210}
i. Work out the acceleration during the first 400 m of the journey.
ii. Find the value of \(T\).
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2. A passenger in a lift has a mass of 84 kg . The lift starts to accelerate at \(1.2 \mathrm {~ms} ^ { - 2 }\). Find the difference between the two possible values of the normal reaction between the lift floor and the passenger.
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3. A particle \(P\) moves along a straight line such that at time \(t\) seconds its velocity \(v \mathrm {~ms} ^ { - 1 }\) is given by: $$v ( t ) = t ^ { 2 } - 5 t + 4$$ Find the distance travelled by the particle between \(t = 1\) and \(t = 5.5\).
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4. A particle of mass \(m \mathrm {~kg}\) is attached to two light inextensible strings \(A C\) and \(B C\). The other ends of the strings are attached to two fixed points \(A\) and \(B\), which are 100 cm apart on a horizontal ceiling. The particle hangs in equilibrium as shown in the diagram, which is not drawn to scale.
\includegraphics[max width=\textwidth, alt={}, center]{6c69f370-0d2d-41ec-8761-0707a6ada43d-08_328_904_301_648} The string \(A C\) has length 80 cm and the string \(B C\) has length 60 cm .
Given that the tension in \(A C\) is 29.4 N , find:
i. the tension in \(B C\)
ii. the value of \(m\).
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5. Two particles \(P\) and \(Q\) have masses 2 kg and 5 kg respectively. The particles are connected by a light inextensible string which passes over a smooth, fixed pulley. Initially both \(P\) and \(Q\) are 2.1 m above horizontal ground. The particles are released from rest with the string taut and the hanging parts of the string vertical, as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{6c69f370-0d2d-41ec-8761-0707a6ada43d-10_417_182_358_1032}
i. Show that the acceleration of \(Q\) as it descends is: \(4.2 \mathrm {~ms} ^ { - 2 }\)
ii. Find the tension in the string as \(Q\) descends.
iii. Explain how you have used the information that the string is inextensible and that the pulley is smooth. When \(Q\) hits the ground it does not rebound and the string becomes slack. Particle \(P\) then moves freely under gravity without reaching the pulley.
iv. Find the greatest height above the ground that \(P\) reaches.
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6. A ski slope is modelled as a rough slope at an angle of \(30 ^ { \circ }\) to the horizontal. A skier of mass 72 kg is being towed up the slope at a constant speed of \(7 \mathrm {~ms} ^ { - 1 }\) by a rope inclined at an angle of \(30 ^ { \circ }\) to the slope. The skier is modelled as a particle \(P\) and the coefficient of friction between the skier and the slope is \(\frac { \sqrt { 3 } } { 23 }\). This situation is represented in the diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{6c69f370-0d2d-41ec-8761-0707a6ada43d-13_396_625_388_790}
i. Show that the value of the normal reaction between the skier and the slope is \(23 \sqrt { 3 } g\) and find a similar expression in terms of \(g\) for the exact value of the tension in the rope.
ii. The skier lets go of the tow rope. Find the time the skier travels for before coming instantaneously to rest, giving your answer as a rational number of seconds.
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