How Does Hooke's Law Relate to Angular Frequency?

[misogynisticfeminist]

hmm ok, i was watching the MIT opencourseware video on oscillations and there was a part where it was mentioned that,

the diff. eq. x''+ \frac {k}{m} x = 0 has solution x= x_0 cos (\omega t + \phi) if and only if \omega= \sqrt{\frac {k}{m}}

how do i show that omega is the sqaureoot of k over m? thanks alot.

 

[Tide]

Differentiate the solution twice with respect to time and substitute it back into the differential equation - then tell us what you discovered! :)

 

[misogynisticfeminist]

ohhh, i see, it can be gotten by verifying the solution. I thought we needed to do something to hooke's law, but verifying the solution works great too.

: )

edit: a tinge of doubt crosses my mind though. When we solve the original differential equation, we get the solution in terms of k, m and x and no omega. While verifying the solution works when the solution is given, and we see that \omega= \sqrt{\frac {k}{m}}. How do we know that \omega= \sqrt{\frac {k}{m}} when we are solving it?

 

[Tide]

If you just substitute x = x_0 \cos(\omega t - \phi) into the differential equation the required value of \omega will jump out at you!

 

[SpaceTiger]

edit: a tinge of doubt crosses my mind though. When we solve the original differential equation, we get the solution in terms of k, m and x and no omega. While verifying the solution works when the solution is given, and we see that \omega= \sqrt{\frac {k}{m}}. How do we know that \omega= \sqrt{\frac {k}{m}} when we are solving it?

In this case, \omega is referring to the angular frequency, which is the multiplicative factor in front of the independent variable in the sine or cosine function. It's just a standard substitution that they probably just didn't bother to define.

 

[DaTario]

try the following:

in your original ODE, consider \omega ^2[\itex] just as a mathematically sound way to express the positiveness of the factor \frac{k}{m} [\itex]. After all the calculations you will realize that this choice has proven itself useful and physically meaningful.