A recovery vessel is salvageing a piece of wreckage from the seabed by pulling it up a steady underwater slope. The slope is inclined at a constant angle β \beta\,β to the horizontal.
The wreckage has a mass of M M\,M kg. It is hauled up the slope by a cable that makes a constant angle θ \theta\,θ with the slope itself. The cable is modeled as light and inextensible, and it remains taut throughout the motion. The wreckage is modeled as a particle.
The coefficient of friction between the wreckage and the seabed is μ\muμ. The wreckage moves up the slope with a constant acceleration a a\,a m s−2^{-2}−2 under a constant tension T T\,T Newtons.
Show that:
T=M(a+gsinβ+μgcosβ)cosθ+μsinθ T = \frac{M(a + g\sin\beta + \mu g\cos\beta)}{\cos\theta + \mu\sin\theta} T=cosθ+μsinθM(a+gsinβ+μgcosβ)In a specific salvage operation, it is found that when M=1200M = 1200M=1200, β=12∘\beta = 12^{\circ}β=12∘, θ=20∘\theta = 20^{\circ}θ=20∘, and T=4500T = 4500T=4500, the wreckage remains stationary. The salvage team attempts to use these values in the formula from part (a), with a=0a = 0a=0, to determine the coefficient of friction μ\muμ. Explain why this method might lead to an incorrect value for μ\muμ.
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.