Mathematical Methods and Mechanics

Question 3

A particle of mass 5 kg is suspended from the ceiling by a model spring of stiffiress 100 Nm-I and natural length 0.5 metres. A model damper with damping constant 50 Nsrrr is attached to the bottom of the particle. The other end of the damper is attached to the floor at a point directly below the point of suspension of the spring. The height of the ceiling above the floor is 2.5 metres. The particle moves in a vertical line. The acceleration due to gravity can be taken as 10 ms''. Take the upward pointing x-a,ris with origin at the floor.

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

(3 marks)

(b) Write down the vector equations of all the forces acting on the particle. (3 marks)

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

i + 10i +20x = 30. (4 marks)

(d) Find the equilibrium position of the particle. (3 marks)

Question 4

Two particles having masses 3 kg a4d 5 kg are moving with velocity vectors i and -2i + i respectively (measured in ms-'). The two particles collide and coalesce to form a composite particle.

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

(4 marks)

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

(c) What percentage of the total kinetic energJ is lost during the collision? (6 marks)