Harmonic oscillator

  • Thread starter quas
  • Start date
  • #1
7
1

Homework Statement


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
שאלה לפורום.JPG

Homework Equations


## \sum F=ma
##
## x(t)=Asin(\omega t + \phi ) ##

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
 

Answers and Replies

  • #2
TSny
Homework Helper
Gold Member
13,157
3,455
##\ddot{x}=g-\frac{k}{m}x ##
OK

##x_0=\frac{c}{k}##
What does c stand for?

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

How do I go from here?
Rewrite the differential equation in terms of y instead of x.
 
  • #3
951
418
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
7
1
OK

What does c stand for?


Rewrite the differential equation in terms of y instead of x.
sorry I meant ## x_0 = \frac{mg}{k}##
ok and then ## \ddot{y}=g-\frac{k}{m}y ## ?
 
Last edited:
  • #5
BvU
Science Advisor
Homework Helper
14,448
3,737
##
\ddot{y}=g-\frac{k}{m}y## can't be right. Compare with ##
\ddot{x}=g-\frac{k}{m}x##
 
  • #6
7
1
##
\ddot{y}=g-\frac{k}{m}y## can't be right. Compare with ##
\ddot{x}=g-\frac{k}{m}x##
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
951
418
##
\ddot{y}=g-\frac{k}{m}y## can't be right. Compare with ##
\ddot{x}=g-\frac{k}{m}x##
??? I must be thick this morning, but if positive y is down that looks ok to me.
 
  • #8
951
418
??? I must be thick this morning, but if positive y is down that looks ok to me.

Oh, good grief. It took me a minute!!
 
  • #9
951
418
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} ##

Much later after it stops oscillating and y is y0 what is the acceleration? g?
 
  • #10
BvU
Science Advisor
Homework Helper
14,448
3,737
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} ##
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: ...:rolleyes:
 

Related Threads on Harmonic oscillator

  • Last Post
Replies
9
Views
2K
  • Last Post
Replies
2
Views
2K
  • Last Post
Replies
3
Views
2K
  • Last Post
Replies
2
Views
1K
  • Last Post
Replies
15
Views
2K
  • Last Post
Replies
4
Views
491
  • Last Post
Replies
13
Views
1K
Replies
3
Views
514
  • Last Post
Replies
6
Views
883
  • Last Post
2
Replies
34
Views
2K
Top