Mathematical Methods and Mechanics

(b) Derive theequation of motion of the particle. (5 marks)

(c) If point B is now stationary, expr€ss the equilibrium position of particle A as measured from o'

(3 marks)

Question 3

(a) Calculate the eigenvalues and its corresponding eigenvectors of a 2x2 matrix grven by

[-+ -,ol [r 7 [

(b) He,nce, find and explain the general equations

(9 marks)

solution of the system of differential

ft = 4x1 -10x2 *z = 3r1 + 7xz.

(3 marks)

Question 4

Two particles, A and B of mass I kg and 2 kg respectively, are attached to a model spnng of natural length lo and stiffiress ,t as shown in Figure Q4. They rest on a rough horizontal table; the coefficient of sliding friction between the particles and the table is p. At time r = 0 the system is at rest with the spring at its natural length and particle ,{ is at the point O. A constant horizontal force of 0.6 N is applied to particle I and at subsequent time r the positions of the particles (measured from thp fixed point O) are x ffidy,as shown. -

A k'1,e, &o.u* --+

v

(a)

Fieure O4

Write down the position of the cenfie of mass of the system, at time / > 0, in terms of the positions of the particles.

(2 marks)

Apply a force diagram for each particle. Express all forces in vector form, giving magnitude and direction

o)