#### Answer

(a) The spring's equilibrium length is 20 cm.
(b) $x = 14 ~cm$
$x = 26 ~cm$
(c) The maximum kinetic energy is 7 joules.
(d) $x = 11.5 ~cm$
$x = 28.5 ~cm$

#### Work Step by Step

(a) At x = 20 cm, $U_s = 0$. Therefore, the spring is neither stretched nor compressed when x = 20 cm. Therefore, the spring's equilibrium length is 20 cm.
(b) The turning points occur when $U_s = TE$. This happens when x = 14 cm and x = 26 cm.
(c) The maximum kinetic energy is equal to the total energy in the system which is 7 joules.
(d) The potential energy in the spring is equal to the total energy in the system when the spring is stretched or compressed by 6 cm. We can find the spring constant as:
$\frac{1}{2}k(0.06~m)^2 = 7~J$
$k = \frac{(2)(7~J)}{(0.06~m)^2}$
$k = 3900~N/m$
If the total energy is doubled, then the total energy is 14 joules. We can find the distance $d$ the string is stretched or compressed when $U_s = TE$
$U_s = TE$
$\frac{1}{2}kd^2 = TE$
$d = \sqrt{\frac{2~TE}{k}}$
$d = \sqrt{\frac{(2)(14~J)}{3900~N/m}}$
$d = 0.085~m = 8.5~cm$
The turning points are x = 11.5 cm and x = 28.5 cm