AC circuit and standing wave questions

[Violagirl]

Homework Statement

1. A 5.0 H inductor is connected to a 120 V rms, 60 Hz ac power supply. Periodically, the energy stored in the inductor reaches a maximum value. What is this maximum value?

A) 0.010 J B) 0.020 J C) 0.40 J D) 0.80 J E) 10.0 J

2. A standing wave on a string is represented by the equation y = A sin (kx) cos (ωt). The distance between consecutive nodes is:

A) ∏/k B) ∏/ω C) 2∏/k D) k/∏ E) ω/∏

The Attempt at a Solution

These were two questions from an exam I had today and I wanted to check to see if my thinking was correct? For the first question, I put C.

I first found the period, 1/f, and got 1/60 = .0167 sec.

Next, I divided the 120 V rms by 5 H and got an answer of 24 V/H. Finally, I multiplied the frequency into this value and got an answer of 0.40 J. Was this correct?

For number 2, I rationalized that a node corresponds to half of a wavelength, which is equivalent to pi. And I know that k is equal the number of wave peaks in a wave. Therefore, out of the choices possible, I thought that A made the most sense. If anyone could confirm whether or not that is correct, I would appreciate it.

 

[NascentOxygen]

For the first question, I put C.

I first found the period, 1/f, and got 1/60 = .0167 sec.

Next, I divided the 120 V rms by 5 H and got an answer of 24 V/H. Finally, I multiplied the frequency into this value and got an answer of 0.40 J. Was this correct?

Hi Violagirl. It sounds like you evaluated the expression V/(f L) ?

That looks suspiciously close to V/(2 Pi f L) which would be the formula for current. You also need a formula relating energy in an inductor to current in it.

For number 2, I rationalized that a node corresponds to half of a wavelength, which is equivalent to pi. And I know that k is equal the number of wave peaks in a wave.

I don't follow that reasoning about k. Take another look at this. First, sketch what the outline (envelope) of the standing wave looks like.

 

[Violagirl]

Thanks for clarifying number 1, it makes more sense now.

For 2, well I know from the equation that k = 2pi/λ. For a standing wave, λ corresponds to 1/2L. And I know it's half a wavelength, it should be equivalent to pi. I guess from there though, I wasn't sure how to understand number 2...

 

[NascentOxygen]

For 2, well I know from the equation that k = 2pi/λ.

Do we? You're talking about the term sin (kx) ?

We know it has a node where kx = 0

 

[Violagirl]

Oooh, true. I guess I was thinking of it in general terms like 0, pi, 2pi, etc. And I guess from a standing wave, since we see one full peak from a standing wave, that I thought that k would correspond to pi since we see it travel from 0 to pi in the distance of a peak. This question was definitely tricky for me to think about.

 

[NascentOxygen]

Oooh, true. I guess I was thinking of it in general terms like 0, pi, 2pi, etc. And I guess from a standing wave, since we see one full peak from a standing wave, that I thought that k would correspond to pi since we see it travel from 0 to pi in the distance of a peak. This question was definitely tricky for me to think about.

I think you mean the product kx would correspond to Pi, 2Pi, 3Pi, etc., don't you?

 

[Violagirl]

Yes, that is true, kx is what would correspond to pi, 2pi, etc.

 

[NascentOxygen]

Yes, that is true, kx is what would correspond to pi, 2pi, etc.

So what is your answer for (2)?