Answer
(a) $k = 430~N/m$
(b) The mass is 4.7 kg
Work Step by Step
(a) When the spring is compressed, the potential energy in the spring is equal to the work $W$ done to compress the spring. Therefore,
$U_s = W$
$\frac{1}{2}kx^2 = W$
$k = \frac{2W}{x^2}$
$k = \frac{(2)(3.6~J)}{(0.13~m)^2}$
$k = 430~N/m$
(b) The mass experiences maximum acceleration when the spring is at maximum compression, because then the spring exerts the strongest force on the mass. We can find the mass $M$ as:
$Ma = kx$
$M = \frac{kx}{a}$
$M = \frac{(430~N/m)(0.13~m)}{12~m/s^2}$
$M = 4.7~kg$
The mass is 4.7 kg.