A heavy-duty cable lift in a mining facility consists of a cargo hopper, AAA, and a solid steel counterweight, BBB, connected by a light inextensible cable. The hopper A A\,A has a mass of 80 kg and is positioned on a rough track inclined at 60∘ 60^\circ\,60∘ to the horizontal. The cable passes over a smooth pulley at the top of the track, and the counterweight BBB, which has a mass of 120 kg, hangs freely.
The system is released from rest with the cable taut and parallel to the line of greatest slope. The counterweight B B\,B descends with acceleration a a\,a m s−2^{-2}−2 and the tension in the cable is T T\,T Newtons.
Explain why the equation of motion for the counterweight B B\,B is 120g−T=120a120g - T = 120a120g−T=120a.
Show that the normal reaction force, RRR, between the hopper A A\,A and the track is 40g Newtons.
The coefficient of friction, μ\muμ, between the hopper A A\,A and the track surface satisfies 0≤μ≤0.50 \le \mu \le 0.50≤μ≤0.5. Show that
a≥(5−23)g10 a \ge \frac{(5 - 2\sqrt{3})g}{10} a≥10(5−23)gState one modelling assumption used in this problem regarding the cable or pulley.
