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.
139 exam-style questions on OCR A Level Maths 3.3 Forces and Newton's Laws, covering 3.3.1 Concept and vector nature of a force, 3.3.2 Newton's first law, 3.3.3 Newton's second law in a straight line, 3.3.4 Newton's second law with vectors (A-level only), 3.3.5 Newton's second law with resolved forces (A-level only), 3.3.6 Weight, 3.3.7 Gravitational acceleration, 3.3.8 Newton's third law, 3.3.9 Normal reaction force, 3.3.10 Smooth contact model, 3.3.11 Equilibrium with connected particles and pulleys, 3.3.12 Newton's third law with resolved forces (A-level only), 3.3.13 Equilibrium by resolving (A-level only), 3.3.14 Equilibrium of forces in two dimensions using vectors (A-level only), 3.3.15 Resolving forces for advanced problems (A-level only), 3.3.16 Resultant forces and components (A-level only), 3.3.17 Dynamics of a particle in a plane (A-level only), 3.3.18 Concept of a frictional force (A-level only), 3.3.19 Contact force components (A-level only), 3.3.20 Coefficient of friction (A-level only), 3.3.21 Static and limiting equilibrium on a rough surface (A-level only), 3.3.22 Motion of a body on a rough surface (A-level only), and 3.3 Forces and Newton's Laws. Each one has a worked solution and a mark scheme showing where the marks go.