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## Homework Statement

a mass m hangs from a uniform spring of spring constant k.

(a)what is the period of oscillations in the system?

(b)what would it be if the mass m were hung so that:

1)it was attached to two identical springs hanging side by side?

2)it was attached to the lower of two identical springs connected end to end?

(P.S. I am not sure if we are to neglect the force of weight of the mass)

## Homework Equations

[itex]T=2\pi /\omega [/itex]

## The Attempt at a Solution

**(a)**[itex]m\ddot{x} +kx=0[/itex]

dividing by "m" you get: [itex]\ddot{x}+\omega^2 x=0[/itex]

to find the roots: [itex]r^2 +\omega^2 = 0[/itex]

to which the roots are: [itex]r=\pm i\omega[/itex]. According to this equation the angular frequency is [itex]\omega[/itex] which equals [itex]\sqrt{k/m}[/itex].

Therefore, [itex]T=2\pi \sqrt{m/k}[/itex]

**(b)(1)**skipping a few steps because its similar to last equation you get [itex]\ddot{x} +\omega^2 x=0[/itex]

again finding the roots you get: [itex]\pm \omega \sqrt{2} i [/itex],

According to this equation the angular frequency is [itex]\sqrt{2}\omega[/itex] which equals [itex]\sqrt{2k/m}[/itex].

the period is: [itex]2\pi \sqrt{m/2k}[/itex]

**(b)(2)**I am not really sure how to set this one up but my thinking was something like this.

I can write the spring force as if it were 1 spring of length 2L.

but the thing is that the oscillations don't depend on the length of the spring, and I am not sure if this is a correct way of going about it. Please help.