Edexcel A Level Further Maths M5 · Mechanics 5 June 2013 question paper 6681R

A particle moves in a plane in such a way that its position vector r metres at time t seconds satisfies the differential equation

When t = 0, the particle is at the origin and is moving with velocity (4i + 2j) m s^–1. Find r in terms of t.

Three forces F₁ = (3i – j + k) N, F₂ = (2i – k) N, and F₃ act on a rigid body. The force F₁ acts through the point with position vector (i + 2j + k) m, the force F₂ acts through the point with position vector (i – 2j) m and the force F₃ acts through the point with position vector (i + j + k) m. Given that the system F₁, F₂ and F₃ reduces to a couple G,

find G.

The line of action of F₃ is changed so that the system F₁, F₂ and F₃ now reduces to a couple (6 i + 8 j + 2 k) N m.

Find an equation of the new line of action of F₃, giving your answer in the form r = a + t b, where a and b are constant vectors.

A spacecraft is moving in a straight line in deep space. The spacecraft moves by ejecting burnt fuel backwards at a constant speed of 2000 m s^–1 relative to the spacecraft. The burnt fuel is ejected at a constant rate of c kg s^–1. At time t seconds the total mass of the spacecraft, including fuel, is m kg and the speed of the spacecraft is v m s^–1.

Show that, while the spacecraft is ejecting burnt fuel,

At time t = 0, the mass of the spacecraft is M₀ kg and the speed of the spacecraft is 2000 m s^–1. When t = 50, the spacecraft is still ejecting burnt fuel and its speed is 6000 m s^–1.

Show, using integration, that the moment of inertia of a uniform solid right circular cone of mass M, height h and base radius a, about an axis through the vertex, parallel to the base, is

[ You may assume without proof that the moment of inertia of a uniform circular disc, of radius r and mass m, about a diameter is mr.]

A uniform circular lamina has radius 2 a and centre C. The points P, Q, R and S on the lamina are the vertices of a square with centre C and CP = a. Four circular discs, each of radius , with centres P, Q, R and S, are removed from the lamina. The remaining lamina forms a template T, as shown in Figure 1. The radius of gyration of T about an axis through C, perpendicular to T, is k.

The template T is free to rotate in a vertical plane about a fixed smooth horizontal axis which is perpendicular to T and passes through a point on its outer rim.

Write down an equation of rotational motion for T and deduce that the period of small oscillations of T about its stable equilibrium position is

A uniform circular disc, of radius r and mass m, is free to rotate in a vertical plane about a fixed smooth horizontal axis L which is perpendicular to the plane of the disc and passes through a point which is from the centre of the disc. The disc is held at rest with its centre vertically above the axis. The disc is then slightly disturbed from its rest position. You may assume without proof that the moment of inertia of the disc about L is

Find the magnitude of the force exerted on the disc by the axis when the disc has