7 One end of a rope is attached to a block A of mass 2 kg . The other end of the rope is attached to a second block B of mass 4 kg . Block A is held at rest on a fixed rough ramp inclined at \(30 ^ { \circ }\) to the horizontal. The rope is taut and passes over a small smooth pulley P which is fixed at the top of the ramp. The part of the rope from A to P is parallel to a line of greatest slope of the ramp. Block B hangs vertically below P , at a distance \(d \mathrm {~m}\) above the ground, as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{41b1f65b-8806-4183-81a1-0276691e203c-05_351_851_1274_244} Block A is more than \(d \mathrm {~m}\) from P . The blocks are released from rest and A moves up the ramp. The coefficient of friction between A and the ramp is \(\frac { 1 } { 2 \sqrt { 3 } }\). The blocks are modelled as particles, the rope is modelled as light and inextensible, and air resistance can be ignored.

1. Determine, in terms of \(g\) and \(d\), the work done against friction as A moves \(d \mathrm {~m}\) up the ramp.
2. Given that the speed of B immediately before it hits the ground is \(1.75 \mathrm {~ms} ^ { - 1 }\), use the work-energy principle to determine the value of \(d\).
3. Suggest one improvement, apart from including air resistance, that could be made to the model to make it more realistic.