Edexcel A Level Further Maths M5 · Mechanics 5 June 2017 question paper 6681

[ In this question, i and j are perpendicular unit vectors in a horizontal, x-y plane. ] A bead P of mass 0.08 kg is threaded on a smooth straight horizontal wire which lies along the line with equation y = 2 x – 1. The unit of length on both axes is the metre. Initially the bead is at rest at the point (a, b). A force (6 i – 2 j) N acts on P and moves it along the wire so that P passes through the point (5, 9) with speed 10 m s^–1. Find the value of a and the value of b.

[ In this question, i and j are perpendicular unit vectors in a horizontal plane and k is a unit vector vertically upwards. ] A particle of mass 2 kg moves under the action of a constant gravitational force –19.6 k N. The particle is subject to a resistive force –v newtons, where v m s^–1 is the velocity of the particle at time t seconds.

By writing down an equation of motion of the particle, show that v satisfies the differential equation

When t = 0, v = (4 i – 6 j + 11.6 k)

Find v when t = ln 4

The position vectors of the points P and Q on a rigid body are (i – 2 j + 3 k)m and (i – j + k)m respectively, relative to a fixed origin O. A force F₁ of magnitude 6 N acts at P in the direction (i – 2j + 2k). A force F₂ of magnitude 14 N acts at Q in the direction (3i – 6j + 2k). When a force F₃ acts at O, which is also a point on the rigid body, the system of three forces is equivalent to a couple of moment G

Find G

When an additional force F₄ = (i + 3j + 4k) N also acts at O, the system of four forces is equivalent to a single force R.

Write down R.

Find an equation of the line of action of R in the form r = a + t b, where a and b are constant vectors and t is a parameter.

A uniform lamina PQR of mass m is in the shape of an isosceles triangle, with PQ = PR = 5 a and QR = 6 a. The midpoint of QR is T.

Show, using integration, that the moment of inertia of the lamina about an axis which passes through P and is parallel to QR, is 8ma².

Show, using integration, that the moment of inertia of the lamina about an axis which passes through P and T, is 1.5ma². [ You may assume without proof that the moment of inertia of a uniform rod, of mass m and length 2l, about an axis perpendicular to the rod through its midpoint is ml₂]

The lamina is now free to rotate in a vertical plane about a fixed smooth horizontal axis A which passes through P and is perpendicular to the plane of the lamina. The lamina makes small oscillations about its position of stable equilibrium.

By writing down an equation of rotational motion for the lamina as it rotates about A, find the approximate period of these small oscillations.

A uniform rod AB, of mass M and length 2 L, is free to rotate in a vertical plane about a smooth fixed horizontal axis through A. The rod is hanging vertically at rest, with B below A, when it is struck at its midpoint by a particle of mass M. Immediately before this impact, the particle is moving with speed u, in a direction which is horizontal and perpendicular to the axis. The particle is brought to rest by the impact and immediately after the impact the rod moves with angular speed .

Immediately after the impact, the magnitude of the vertical component of the force exerted on the rod at A by the axis is

Find u in terms of L and g.

Show that the magnitude of the horizontal component of the force exerted on the rod at A by the axis, immediately after the impact, is zero.

The rod first comes to instantaneous rest after it has turned through an angle

Find the size of

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A small object P, of mass m₀, is projected vertically upwards from the ground with speed U. As P moves upwards it picks up droplets of moisture from the atmosphere. The droplets are at rest immediately before they are picked up. In a model of the motion, P is modelled as a particle, air resistance is assumed to be negligible and the acceleration due to gravity is assumed to have the constant value of g. When P is at a height x above the ground, the combined mass of P and the moisture is m₀(1 + kx), where k is a constant, and the speed of P is v.

Show that, while P is moving upwards

The general solution of this differential equation is given by v² = − 1 + kx, where A is an arbitrary constant.

find, in terms of h, the height of P above the ground when P first comes to rest.