Question

HELPU!
Per definition; cos(w' 2T) is always equal to one.
Why? There is no explanation to why it will always be one. How can w'2T always result into one? ( Attachment )

Answer 1

I asked the same question here ( http://openstudy.com/study#/updates/55e571ede4b03567e110d72b )
and got an answer; but realized that w is derived! How can a derived w' cancel out a w?

Answer 2

Ok well I looked at the question you asked and that was short and sweet and makes perfect sense so there's not much I can really say about that. One thing though that might help though is this:

\[\cos(x) = \cos( x+ 2 \pi)\]
But we could do this any number of times, so really:
\[\cos(x)=\cos(x+n2 \pi)\]
If we evaluate this at x=0 what do we have?

\[\cos(0)=\cos(n 2 \pi)\]\[1=\cos(n 2 \pi)\]

So it looks like here in your problem you have n=2 possibly?

\[1=\cos(4 \pi)\]

Answer 3

@Kainui
I understand that; but not in this case, when we have a w' 2pi inside. As w can be different; we dont really know it will always be result into 1.

Now if w was not w', it would make perfect sense mathematically; as it would just become n2pi, as you mentioned. However w' cannot cancel out w (I believe), so it does not mathematically make sense

Answer 4

I don't know what the difference is between \(\omega\) and \(\omega'\) is since I don't know where they came from.

Answer 5

Like I am particularly suspicious that you might be blindly using formulas that apply to one certain case and trying to apply them generally.

Answer 6

w = root(k/m)

w' = root(k/m - b^2/4m^2)

Their values should not matter, as, according to previous answer (link in previous post) they should cancel eachother out; resulting into an cos(n*2pi) value resulting into it always becoming = 1

Answer 7

(or rather; the values of k, m, and so on) - the variables matter of course!

Answer 8

Where does \(\frac{-b^2}{4m^2}\) come from?

Answer 9

No idea - Its from a formula book

Answer 10

There's your problem

Answer 11

It might be dampening, hmm

Answer 12

Thanks for helping! Ill ask the teacher if I can get a hold of him :)

Answer 13

Yeah for all you know \(\omega = \omega'\)

Answer 14

That is to say:

\[T = \frac{2 \pi}{\omega'}\]

Is totally possible, you don't know.

Answer 15

Then it would make perfect sense; yes :) Ill look further into it

Answer 16

if the confusion is over w and w',

w' isnt the derivative of w,
w and w' are used to represent undamped natural frequency and damped frequency

Answer 17

\[T=2 \pi \sqrt{ \frac{m}{k}}\]
this is the time period for the undamped system, you cannot substitute it here since you have damping 'b'