Body on a spring: expression for the period T

[moenste]

Homework Statement

(a) A body of mass m is suspended from a vertical, light, helical spring of force constant k, as in Fig. 1. Write down an expression for the period T of vertical oscillations of m.

(b) Two such identical springs are now joined as in Fig. 2 and support the same mass m. In terms of T, what is the period of vertical oscillations in this case?

(c) The identical springs are now placed side by side as in Fig. 3, and m is supported symmetrically from them by means of a weightless bar. In terms of T, what is the period of vertical oscillations in this case?

2. The attempt at a solution
(a) The first figure looks like a basic example of a body on a spring and it's period should be: T = 2 π √(m / k).

(b) Since there are two identical springs, I would say that their force constant k should be less, because the spring is longer now: T = 2 π √(m / (k / 2)).

(c) Since the two springs are in parallel, the force constant k should be twice as much: T = 2 π √(m / (2k)).

I am not sure for the (b) and (c) parts. If we take the (a) formula as the basis, the only thing (as I see it) that changes is k due to the different spring positions. So, for (b) it should be (k / 2) and for (c) 2k. I am going in the right direction?

 

[Doc Al]

If we take the (a) formula as the basis, the only thing (as I see it) that changes is k due to the different spring positions. So, for (b) it should be (k / 2) and for (c) 2k. I am going in the right direction?

Exactly right!