# Harmonic oscillator

1. Dec 26, 2016

### quas

1. The problem statement, all variables and given/known data
a mass is placed on a loose spring and connected to the ceiling. the spring is connected to the floor in t=0 the wire is cut
a. find the equation of the motion
b. solve the equation under the initial conditions due to the question

2. Relevant equations
$\sum F=ma$
$x(t)=Asin(\omega t + \phi )$
3. The attempt at a solution
a. due to the 2 law of newton: $\sum F=ma_x$
$mg-kx=ma$
$mg-kx=m\ddot{x} \\ \ddot{x}=g-\frac{k}{m}x$

b. first I'll find the point equilibrium
$c-kx=ma \\ c-kx=m\cdot 0 \\ x_0=\frac{c}{k}$

then I'll define $y=x-x_0$

How do I go from here?
thanks

2. Dec 26, 2016

### TSny

OK

What does c stand for?

Rewrite the differential equation in terms of y instead of x.

3. Dec 26, 2016

### Cutter Ketch

As TSny hints, you tripped finding x0. Think again what condition will be true at equilibrium and define x0 again. There won't be an unknown c.

4. Dec 28, 2016

### quas

sorry I meant $x_0 = \frac{mg}{k}$
ok and then $\ddot{y}=g-\frac{k}{m}y$ ?

Last edited: Dec 28, 2016
5. Dec 28, 2016

### BvU

$\ddot{y}=g-\frac{k}{m}y$ can't be right. Compare with $\ddot{x}=g-\frac{k}{m}x$

6. Dec 28, 2016

### quas

I might have a barrier but I do not understand how to build a differential equation for y,,,,
I understand that $\dot{x}=\frac{dx}{dt}=\frac{dy}{dt}=\dot{y} \\ a=\ddot{x}=\frac{d^2x}{dt^2}=\frac{d^2y}{dt^2}=\ddot{y}$

7. Dec 28, 2016

### Cutter Ketch

??? I must be thick this morning, but if positive y is down that looks ok to me.

8. Dec 28, 2016

### Cutter Ketch

Oh, good grief. It took me a minute!!

9. Dec 28, 2016

### Cutter Ketch

Much later after it stops oscillating and y is y0 what is the acceleration? g?

10. Dec 28, 2016

### BvU

Simple: the second derivatives on the left are equal alright. But you need to substitute your expression for y in terms of x. Not just y = x, but: ...