spring length calculations?

Question

A 6.80 kg mass suspended from a spring with spring constant, k =400 N/m, extends it to a total length of 0.290 m. Find the new total length of the spring when a 15.60 kg mass is suspended

Answer 1

In order to find the new total length, we have to establish the relaxed length of the spring (when no mass is suspended from it). We write:
L1 = Lo + x1 and L2 = Lo + x2
where Lo = relaxed length and
x1 and x2 are the tension displacement, which can be written using Hooke's law:
x1 = Fg1/k and x2 = Fg2/k, where Fg1 and Fg2 are the weights and k is the constant of the spring.
From the first equation we write Lo = L1 ─ x1 and substitute this into the second equation:
L2 = L1 ─ x1 + x2 = L1 + x2 ─ x1 = L1 + (Fg2 ─ Fg1) / k = 0.290 m + (15.60 ─ 6.80)x10 N / 400 Nm-1 = 0.510 m

Answer 2

You're mixing several aspects of this type of problem. 1) The tension in both springs combine is equal to the 40N weight. Since the springs are identical and are stretching the same amount, the springs are symmetrical about the center of gravity of the weight, the tension in each spring is 1/2 the 40N weight. 2) The spring constant k can be found from the tension in each spring k=F/delta L where F=1/2*40N (weight)= tension in one spring