Edexcel A Level Further Maths M5 · Mechanics 5 June 2016 question paper 6681

[ In this question, i and j are perpendicular unit vectors in a horizontal plane. ] A bead P of mass 0.4 kg is threaded on a smooth straight horizontal wire. The wire lies along the line with vector equation r = (i + 2j) + (–2i + 3j). The bead is initially at rest at the point A with position vector (– i + 5 j) m. A constant horizontal force (0.5 i + j) N acts on P and moves it along the wire to the point B. At B the speed of P is 5 m s^–1. Find the position vector of B.

A particle P is moving in a plane. At time t seconds the position vector of P is r metres and the velocity of P is v m s^–1. When t = P is instantaneously at rest at the point with position vector (i – j) m. Given that r satisfies the differential equation

find v in terms of t.

Three forces F₁, F₂ and F₃ act on a rigid body at the points with position vectors r₁, r₂ and r₃ respectively, where F₁ = (2j – k) N F₂ = (i + k) N F₃ = (i + j) N r₁ = (4j – k) m r₂ = (2i + k) m r₃ = (3i + j + k) m The system of the three forces is equivalent to a single force R acting through the point with position vector (i – j + k) m, together with a couple of moment G.

Find R.

Find G.

Find, using integration, the moment of inertia of a uniform cylindrical shell of radius r, height h and mass M, about a diameter of one end.

A uniform piece of wire ABC, of mass 2 m and length 4 a, is bent into two straight equal portions, AB and BC, which are at right angles to each other, as shown in Figure 1. The wire rotates freely in a vertical plane about a fixed smooth horizontal axis L which passes through A and is perpendicular to the plane of the wire.

Show that the moment of inertia of the wire about L is

By writing down an equation of rotational motion for the wire as it rotates about L, find the period of small oscillations of the wire about its position of stable equilibrium.

A firework rocket, excluding its fuel, has mass m₀ kg. The rocket moves vertically upwards by ejecting burnt fuel vertically downwards with constant speed u m s^–1, u > 24.5, relative to the rocket. The rocket starts from rest on the ground at time t = 0. At time t seconds, t 2, the speed of the rocket is v m s^–1 and the mass of the rocket including its fuel is m₀(5 – 2t) kg. It is assumed that air resistance is negligible and the acceleration due to gravity is constant.

Show that, for t 2

Find the speed of the rocket at the instant when all of its fuel has been burnt.

A uniform square lamina PQRS, of mass m and side 2 a, is free to rotate about a fixed smooth horizontal axis which passes through P and Q. The lamina hangs at rest in a vertical plane with SR below PQ and is given a horizontal impulse of magnitude J at the midpoint of SR. The impulse is perpendicular to SR.

Find the initial angular speed of the lamina.

Find the magnitude of the angular deceleration of the lamina at the instant when the lamina has turned through radians.

Find the magnitude of the component of the force exerted on the lamina by the axis, in a direction perpendicular to the lamina, at the instant when the lamina has turned