SPS SPS SM Mechanics (SPS SM Mechanics) 2021 May

1.
\includegraphics[max width=\textwidth, alt={}, center]{14759a7d-dbec-4d72-a722-c83043fb59c5-04_607_894_150_644} Three horizontal forces of magnitudes \(F \mathrm {~N} , 8 \mathrm {~N}\) and 10 N act at a point and are in equilibrium. The \(F \mathrm {~N}\) and 8 N forces are perpendicular to each other, and the 10 N force acts at an obtuse angle \(( 90 + \alpha ) ^ { \circ }\) to the \(F \mathrm {~N}\) force (see diagram). Calculate

1. \(\alpha\),
2. \(F\).

2. A particle \(P\) moves with constant acceleration \(( 3 \mathbf { i } - 5 \mathbf { j } ) \mathrm { ms } ^ { - 2 }\). At time \(t = 0\) seconds \(P\) is at the origin. At time \(t = 4\) seconds \(P\) has velocity \(( 2 \mathbf { i } + 4 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\).

1. Find the displacement vector of \(P\) at time \(t = 4\) seconds.
2. Find the speed of \(P\) at time \(t = 0\) seconds.

3. A uniform plank \(A B\) has weight 100 N and length 4 m . The plank rests horizontally in equilibrium on two smooth supports \(C\) and \(D\), where \(A C = x \mathrm {~m}\) and \(C D = 0.5 \mathrm {~m}\) (see diagram).
\includegraphics[max width=\textwidth, alt={}, center]{14759a7d-dbec-4d72-a722-c83043fb59c5-08_184_1266_283_402} The magnitude of the reaction of the support on the plank at \(C\) is 75 N . Modelling the plank as a rigid rod, find

1. the magnitude of the reaction of the support on the plank at \(D\),
2. the value of \(x\). A stone block, which is modelled as a particle, is now placed at the end of the plank at \(B\) and the plank is on the point of tilting about \(D\).
3. Find the weight of the stone block.
4. Explain the limitation of modelling
   (a) the stone block as a particle,
   (b) the plank as a rigid rod.

4. A particle \(P\) projected from a point \(O\) on horizontal ground hits the ground after 2.4 seconds. The horizontal component of the initial velocity of \(P\) is \(\frac { 5 } { 3 } d \mathrm {~ms} ^ { - 1 }\).

1. Find, in terms of \(d\), the horizontal distance of \(P\) from \(O\) when it hits the ground.
2. Find the vertical component of the initial velocity of \(P\).
   \(P\) just clears a vertical wall which is situated at a horizontal distance \(d \mathrm {~m}\) from \(O\).
3. Find the height of the wall. The speed of \(P\) as it passes over the wall is \(16 \mathrm {~ms} ^ { - 1 }\).
4. Find the value of \(d\) correct to 3 significant figures.

5.
\includegraphics[max width=\textwidth, alt={}, center]{14759a7d-dbec-4d72-a722-c83043fb59c5-12_501_880_132_614} Two particles \(P\) and \(Q\) are attached to opposite ends of a light inextensible string which passes over a small smooth pulley at the top of a rough plane inclined at \(30 ^ { \circ }\) to the horizontal. \(P\) has mass 0.2 kg and is held at rest on the plane. \(Q\) has mass 0.2 kg and hangs freely. The string is taut (see diagram). The coefficient of friction between \(P\) and the plane is 0.4 . The particle \(P\) is released.

1. State the tension in the string before \(P\) is released, and find the tension in the string after \(P\) is released.
   \(Q\) strikes the floor and remains at rest. \(P\) continues to move up the plane for a further distance of 0.8 m before it comes to rest. \(P\) does not reach the pulley.
2. Find the speed of the particles immediately before \(Q\) strikes the floor.
3. Calculate the magnitude of the contact force exerted on \(P\) by the plane while \(P\) is in motion. End of Examination Extra Answer Space