Mathematical Methods and Mechanics

Question 3

A particle of mass rz is connected to the left-hand wall by a model spring of stiffiress ft and natural length /0, and a model damper with damping constant r, and on the right by another model damper of damping constant r. The position of the particle at time / is x (measured fromthe left-hand wall).

tr'ieure O3

The vibrating system is oscillating horizontally as shown above, with symbols in their usual meaning. The forcing functionflr) is on the displacement at B.

(a) Draw a diagram clearly indicating the forces acting on the particle and the positive x- direction.

(3 marks)

(b) Write down the vector equations of all the forces acting on the particle. (4 marks)

(c) Show that the equation of motion ofthe particle is given by,

mi+2r*+kx=klo+r!. (6 marks)

Question 4

Two identical particles, each having a mass of 2 kg are moving with velocities -4i + 3j and 2i + 5j respectively (measured in ms-1). The two particles collide and coalesce to form a composite particle.

(a) Using the conseryation of linear momentum, write down the vector equation which will determine the velocity of the composite particle after the collision.

(3 marks)

(b) Find the final velocity of the composite particle. (3 marks)

(c) Perform calculations to determine if the collision is elastic. (7 marks)