A researcher is hauling a seismic sensor of mass mmm kg up a rough Martian slope inclined at an angle θ\thetaθ to the horizontal. The sensor is pulled by a light, inextensible cable which makes a constant acute angle ϕ\phiϕ with the slope. The cable remains taut as the sensor is hauled with a constant acceleration a m s−2a \text{ m s}^{-2}a m s−2 up the slope. The coefficient of friction between the sensor and the slope is μ\muμ. The tension in the cable is PPP Newtons.
Show that:
P=m(a+gsinθ+μgcosθ)cosϕ+μsinϕ P = \frac{m(a + g \sin \theta + \mu g \cos \theta)}{\cos \phi + \mu \sin \phi} P=cosϕ+μsinϕm(a+gsinθ+μgcosθ)In a trial run where m=12m = 12m=12, θ=20∘\theta = 20^{\circ}θ=20∘, ϕ=15∘\phi = 15^{\circ}ϕ=15∘, and P=85P = 85P=85, the sensor is found to remain stationary. The researcher uses these values in the formula from part (a) to estimate a value for μ\muμ. Explain why this estimate of μ\muμ may be incorrect.
129 exam-style questions on Edexcel A Level Maths Applications of Forces, covering 7.1 Static Particles, 7.2 Modelling with Statics, 7.3 Friction and Static Particles, 7.4 Static Rigid Bodies, 7.5 Dynamics and Inclined Planes, and 7.6 Connected Particles 2. Each one has a worked solution and a mark scheme showing where the marks go.