Period of a Mass-Spring System Ranking Task?

Question

Different mass crates are placed on top of springs of uncompressed length L_0 and stiffness k. The crates are released and the springs compress to a length L before bringing the crates back up to their original positions.

Rank the time required for the crates to return to their initial positions from largest to smallest.
Rank from largest to smallest.
1 )
k= 10 n/m
L= 5 cm
L_0= 10 cm

2)
k= 5 n/m
L= 10 cm
L_0= 20 cm

3)
k= 20 n/m
L= 5 cm
L_0= 15 cm

4)
k= 15 n/m
L= 10 cm
L_0= 15 cm

5)
k= 10 n/m
L= 5 cm
L_0= 20 cm

6)
k= 5 n/m
L= 5 cm
L_0= 10 cm

Answer 1

For each example, find the mass of the crate. The mass of the crate will be m such that kx = mg (an equilibrium condition), where x = L_0 - L and g = 9.8 m/s^2. Don't forget to convert x into meters. The period of a mass-spring system is proportional to sqrt(m/k), so the order of systems from longest to least time for recovery will be the same as the order of decreasing m/k, and you will be able to calculate m/k for each system.