M1 June 2017 Q8

EdexcelOld spec15 marksConnected ParticlesForces & Newton's Laws

8.

Figure 3: particle A (2m) on a horizontal table at distance d from a pulley at P, joined by a string to particle B (m) hanging at height h above the floor
Figure 3

Two particles, \(A\) and \(B\), have masses \(2m\) and \(m\) respectively. The particles are attached to the ends of a light inextensible string. Particle \(A\) is held at rest on a fixed rough horizontal table at a distance \(d\) from a small smooth light pulley which is fixed at the edge of the table at the point \(P\). The coefficient of friction between \(A\) and the table is \(\mu\), where \(\mu < \dfrac{1}{2}\). The string is parallel to the table from \(A\) to \(P\) and passes over the pulley. Particle \(B\) hangs freely at rest vertically below \(P\) with the string taut and at a height \(h\), \((h < d)\), above a horizontal floor, as shown in Figure 3. Particle \(A\) is released from rest with the string taut and slides along the table.

(a)
(i) Write down an equation of motion for \(A\).
(ii) Write down an equation of motion for \(B\). (4)
(b) Hence show that, until \(B\) hits the floor, the acceleration of \(A\) is \(\dfrac{g}{3}(1 - 2\mu)\). (3)
(c) Find, in terms of \(g\), \(h\) and \(\mu\), the speed of \(A\) at the instant when \(B\) hits the floor. (2)

After \(B\) hits the floor, \(A\) continues to slide along the table. Given that \(\mu = \dfrac{1}{3}\) and that \(A\) comes to rest at \(P\),

(d) find \(d\) in terms of \(h\). (5)
(e) Describe what would happen if \(\mu = \dfrac{1}{2}\) (1)
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