A particle of mass m is attached to a fixed point A by a model spring of natural length l_o and stiffness mg/l_o, and to another fixed point B vertically below A by another model spring, of natural length l_o and stiffness 3mg/l_o, where g is the acceleration due to gravity. The distance between point A and B is 3k. The particle moves along the vertical line between A and B as shown in Figure.

(i) Show that the differential equation for the motion of the particle relative to the upper fixed point A is given by x+(4g/l_o)x= 8g

(ii) Determine the general solution of this differential equation.

(iii) Write down the period of oscillations of the particle, and its equilibrium position relative to A.

(iv) Suppose that, when the particle is at its equilibrium position, it is given a blow so that its speed instantaneously becomes v_o in the downward direction. The particle is observed just to reach the fixed point B in its subsequent motion. Use your answer to (ii) and (iii) to find the value of vo in terms of l_o and g.

(v) Find an expression for the total mechanical energy E of the particle when it is a distance x below A and is moving with instantaneous speed v. Take the point A as the datum for potential energy.

(vi) Use the initial condition in (iv) to determine E in terms of vo, m, l_o and g and hence write E in terms of m, g and to only. Then use your result, together with your answer to (v), to find the distances below A at which the particle is instantaneously at rest.