1
$\begingroup$

I was wondering if maybe someone could look at this excerpt from a textbook (Attached). It states that “the displacement of the two masses will be in opposite directions (out of phase by pi)” but I thought that the Definition of Phase was: Phase = theta in the following formula, cos(wt+theta).

However, the two equations in this book excerpt have the exact same expression in parenthesis, and both are nested in the same function, cosine - so how is it that they have different phases? Many thanksSee image of book example attached.

Improve this question
$\endgroup$
1
$\begingroup$

$$-4\cos{(\sqrt{5}t)} = +4\cos{(\sqrt{5}t + \pi)}$$

Improve this answer
$\endgroup$
3
  • $\begingroup$ That’s interesting and very helpful! But I’m interested to know why this works? If we were to graph both equations (y=4cos(sqrt(5)t) and y=-4cps(sqrt(5)t), we’d find that the negative one was simply reflected over the t axis, yes? So then, why the phase shift of pi? It doesn’t look like a phase shift (as in one has y=0 at origin vs. another one at y=pi)? $\endgroup$ – Yelena Dec 4 '20 at 19:33
  • $\begingroup$ @Yelena being "reflected" is one way to see it, but how about you "shift" the second equation until it exactly overlaps the first: now, see how far you had to shift it in order for this to happen. $\endgroup$ – Philip Dec 4 '20 at 19:35
  • $\begingroup$ Thanks so much, that is so interesting! $\endgroup$ – Yelena Dec 4 '20 at 19:37
3
$\begingroup$

Here's a simple GIF to complement @garyp's succinct answer:

enter image description here

I've plotted the two functions $-\cos{(\sqrt{5}t)}$ (blue) and $\cos{(\sqrt{5}t + \phi)}$ (orange), and I've varied $\phi$ between 0 and $2\pi$. As you can see, they start off being exactly exactly reflected, as you'd expect, and now as I tune $\phi$, the orange curve moves "closer" to the blue one until it matches it exactly when $\phi = \pi/2$, as one would expect.

Improve this answer
$\endgroup$
6
  • $\begingroup$ I was expecting an animation when you mentioned "GIF". Oh well... $\endgroup$ – wyphan Dec 4 '20 at 20:07
  • 1
    $\begingroup$ @wyphan It's animated for me! Doesn't it work for you? :'( $\endgroup$ – Philip Dec 4 '20 at 20:17
  • $\begingroup$ Oh, it works now! Must be my ISP messing around with me again... $\endgroup$ – wyphan Dec 4 '20 at 20:18
  • 1
    $\begingroup$ @wyphan Ok it turns out that the GIF I made didn't loop, so you basically see it once and then you don't see it again for a while :P Just made one that loops, so it should be better! :) $\endgroup$ – Philip Dec 4 '20 at 20:22
  • $\begingroup$ Wow thank you so much - what an awesome animation! What programming did you use to do that? $\endgroup$ – Yelena Dec 4 '20 at 21:35

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.