## Physics for Scientists and Engineers: A Strategic Approach with Modern Physics (3rd Edition)

(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) $x = 11.5 ~cm$ $x = 28.5 ~cm$
(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