A particle, of mass \( 9 \mathrm{~kg} \), is attached to two identical springs. The other ends of the springs are attached to fixed points, \( A \) and \( B \), which are \( 1.2 \) metres apart on a smooth horizontal surface. The springs have modulus of elasticity \( 45 \mathrm{~N} \) and natural length \( 0.4 \mathrm{~m} \).
The particle is released from rest at a distance of \( 0.5 \) metres from \( B \) and moves on the line \( A B \). The midpoint of \( A B \) is \( C \). At time \( t \) seconds after release, the displacement of the particle from \( C \) is \( x \) metres, where the direction from \( A \) to \( B \) is taken to be positive.
(a) Show that the resultant force on the particle, at time \( t \), is \( -225 x \) newtons.
(b) Hence show that the particle moves with simple harmonic motion.
(c) State the period of this motion.
(d) Find the speed of the particle when it is \( 0.05 \) metres from \( C \).
(e) Write down an expression for \( x \) in terms of \( t \).
