# Period of oscillation with time independent factor

How can you determine the period of oscillation from a mass that is suspended from the ceiling? The equation becomes: ${{d^2x}\over dt^2}+kx-mg=0$. I am confused by the constant $mg$, because in the classical damped harmonic motion, the damping is velocity dependent.

• $kx - mg = k(x - mg/k)$ – Phoenix87 Jan 14 '15 at 16:05
• There is no damping in a mass that is suspended from the ceiling, unless you take in account air resistance. – Mark Fantini Jan 14 '15 at 16:54

This actually appears to be "ye olde harmonic oscillator" that has been shifted by a constant amount and not a damped harmonic oscillator that you state it is.

To see the shifted behavior, let $$\eta(t)=x(t)-\frac{mg}{k}$$ Direct substitution gives you $$\frac{d^2\eta}{dt^2}+k\eta=0\tag{1}$$ where we used that $mg/k$ is constant for the time-derivative term. (1) is your standard harmonic oscillator.

• This is very helpful. – Michael Angelo Jan 14 '15 at 18:51