What Phase Difference Causes Maximum SHM Separation Equal to Amplitude?

[manjuvenamma]

Two particles are executing SHM with same amplitude and frequency (but with a phase difference). The maximum separation between them was found to be equal to the amplitude. What is the phase difference. This is a question that I came across.

And I solved it the following way.

Assume one is y1 = A sin (wt) and the other one y2 = A sin (wt + phi)
max(y1-y2) = A (given)
Solve for phi.

Since both y1 and y2 are sinusoids of w frequency, their difference will also be a sinusoid with a different amplitude but with same frequency and a different phase.

Since we are just interested in the max value of the difference, I calculated the amplitude of the sinusoid using phasors A e^j(phi) - A. It turned out to be 2A sin(phi/2). From that I calculated phi as pi/3 but the answer I have is as pi/2. I think the book is correct but I could not find the mistake in my calculation.

Can some one point out if I am wrong and if yes, where?

 

[Shooting Star]

The correct answer is pi/3. The answer in the book is wrong.

 

[manjuvenamma]

I think I am wrong because the difference between A sin(wt) and A sin (wt+pi/2) is A at t=0 itself. I think the difference between A sin(wt) and A sin (wt+pi/3) is never A ( which is what is required). I am not able to find a mistake in my method though. Can you tell at what 't' the the difference between A sin(wt) and A sin (wt+pi/3) is A? Thanks for your time and effort.

 

[alphysicist]

Hi manjuvenamma,

You can't find a specific time in seconds until you specify $\omega$, but here is one time that works:

$$t = \frac{1}{\omega} (2.61799\ \mbox{radians})$$

(So if you just use $\omega t=2.61799$ radians as a whole into your functions you will get the right answer.)

You can find this by solving (after cancelling out the A's) for $t$:

$$\sin(\omega t +\pi/3) -\sin(\omega t) = \pm 1$$

 

[Shooting Star]

I think I am wrong because the difference between A sin(wt) and A sin (wt+pi/2) is A at t=0 itself. I think the difference between A sin(wt) and A sin (wt+pi/3) is never A ( which is what is required). I am not able to find a mistake in my method though. Can you tell at what 't' the the difference between A sin(wt) and A sin (wt+pi/3) is A? Thanks for your time and effort.

Yes, at first glance it may seem so. When particle 1 is at x=0, and particle 2 is at x=A, then their separation is A. But the problem states that the maximum separation is A, which is not true here..

Let us analyze this situation stated above.

Case 1:

2 is at x=A, and 1 is going away from x=0 in the -ve direction. Since the speed of 2 at the extreme point is very small compared to 1, without any calculation we can say that their separation will be more than A for some time after this.

Case 2:

2 is at x=A, and 1 is going toward 2 from x=0, that is, in the +ve direction. We know that when 1 will reach the extreme position, 2 would have come to the mean position x=0. After that, the same argument as in case 1 applies.

As we see, in both of these cases, the maximum separation is more than A.

What you have written in the first post is correct, and the answer is pi/3.

 

[manjuvenamma]

Thanks so much, Shooting Star and alphysicist!
I am now convinced that I am right and I know the correct method too!