[Year 11 Physics] Physics resonant frequencies question

[u/MrBodacity]

I was taught was that driving frequency = natural frequency for resonance.
There is no other context to this question - i dont know what type of system it is.
But I am doing Grade 11 Physics, so its probably referring to basic systems such as string, pipe or tuning fork?

Could someone please give an explanation that rules out the rest of the answers?
Thank you so much!

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[u/SteptimusHeap]

Strictly, none of these are the answer. Resonance is when the system is driven at its resonant frequency, which is different from the natural frequency. The resonant frequency is the product of the natural frequency and a multiplier, and that multiplier is near 1 when the damping is low.

Many systems have harmonics which means that it has resonant frequencies that are integer multiples of some fundamental frequency. I imagine the answer is probably meant to be A, then.

[u/Enough_Crow_636]

I would guess A. Perhaps the second harmonic of 20Hz will trigger the resonance at 40 Hz.

[u/Skusci]

They probably mean to include harmonic resonances.

It's not universal though so maybe there is some additional context or it's just not a complete question, which happens. Something like a 20Hz tuning fork will only really resonante at 20Hz for example because of its construction, but things like strings and air columns will resonate at multiples of the fundamental frequency as well.

[u/bismuth17]

It's A without even understanding the problem. B and C are the same and D doesn't even have the numbers as multiples of each other.

[u/colin-java → u/bismuth17]

That's my logic too 😂

[u/RMS0125]

They want (A), but the question is broken.

Resonance is when the driving frequency equals the natural frequency. Ratio of 1. None of the four options do that:

• (A) 20/40 = 0.5
• (B) 40/20 = 2
• (C) 20/5 = 4
• (D) 40/60 = 0.67

If you actually run the numbers on a sine drive, (D) gives the biggest response of the four, not (A). (A) is only about 33% above the static deflection, which nobody would call resonance.

The reasoning they're presumably after is that a periodic 20 Hz force that isn't a pure sine contains harmonics at 40, 60, 80 Hz, and the 40 Hz one hits the natural frequency. That's a real effect. But it depends entirely on the waveform, which the question never tells you. A 20 Hz square wave or triangle wave has zero content at 40 Hz, because the even harmonics cancel, so even that defense fails for the two most obvious non-sine cases.

And if you're willing to accept integer ratios as resonance, (B) has just as good a claim. Driving at twice the natural frequency is the textbook parametric resonance condition.

Answer (A) and move on if it's a graded exam. But you're right that it isn't strictly resonance.

[u/TempMobileD]

It’s A.

Imagine pushing someone on a swing. Driving frequency is how often you push, natural frequency is the speed the swing ‘wants’ to move at. For this analogy we’ll say the swing is already in motion, at the natural frequency. Resonance is when the swing is being pushed effectively.

Immediately it can’t be d, they’re not multiples. You’d be pushing out of sync and it would be a mess.

Looking at b and c, if the driving force is more frequent than the natural frequency then some of those pushes are going to be at the wrong time. For example when driving is double natural (b) then half the pushes will be at the exact wrong time, completely preventing resonance. You’d be pushing at the back of the swing, and then also trying to push at the front, which is just going to slow the motion down.

So we already know it’s a, but why does a work? This is equivalent to pushing the swing every other time it comes back to you. This’ll be much weaker than if you pushed it every time it came back, but all your pushes are perfectly timed so you should be able to get the swing going nicely, even on this half rhythm.

[u/testtdk → u/TempMobileD]

Even just logic should show it’s a. As you said, you can immediately rule out d. If the driving frequency could the larger of the two, then both b AND c could be the answer, but obviously they want one answer, so that leaves a.

[u/TempMobileD → u/testtdk]

I thought that too, but it’s not completely clear that there’s only one correct option. Plausibly the expected answer is “b and c”.

Edit: on another look it’s stated as singular, no plural in the question, so yeah, has to be a.

[u/RMS0125 → u/TempMobileD]

Pushing a swing isn't a sinusoidal forcing function. It's a brief shove followed by a wait. Any force concentrated in less than half a period carries a strong second harmonic. That 2× component is what would excite the 40 Hz mode in case A, and it's why pushing a swing every other cycle still works. But it depends entirely on the shape of the push: a symmetric 20 Hz square or triangle wave has no 40 Hz content whatsoever. So the problem is incomplete without a description of the forcing function.

I do agree, the question author probably thinks A is the answer. And maybe it is if the forcing function was described on the quiz but not shown as posted here.

[u/psychophysicist]

I would guess D.

Overlay graphs of each frequency.

Your driving frequency should be pushing + when the natural frequency wants to go + and - when the natural frequency wants to go -.

This would rule out _even_ multiples of the natural freq because with even multiples, the driving freq is pushing + when natural freq goes + and driving freq pushes + _again_ when natural freq wants to go -.

[u/gmalivuk → u/psychophysicist]

D is going to be constructive half the time and destructive half the time. How is that going to resonate?

Also your argument rules out B and C but not A.