BJ0001 : Four coupled pendulums and finding which one has the maximum amplitude

John Rennie

Jun 10, 2020 08:22

Bhavay

Are there methods(that i am not aware about) that will make calculation faster, for ex i read Lang. Mechanics make newtonian mechanics questions easy?

John Rennie

Honestly, it's just experience. When you're first starting out everything seems impossibly hard but the more problems you do the quicker you'll get at doing them.

Lagrangian mechanics is too complicated for JEE level stuff. You don't want to get involved with it - trust me on this one :-)

Bhavay

I 100% trust u. :-).Let's start the next one without further ado , shall we?

John Rennie

To be honest I'm not sure how to answer this because I'm not sure I understand what it's asking. I would guess that the elastic string at the top transfers energy between the pendulums so they all have an equal energy.

Bhavay

The answer given is B

is it related to domino affect?

John Rennie

Jun 10, 2020 08:28

OK, then I'm wrong because if they have equal energy D would have the largest amplitude. In that case I'm afraid I have no ideahow this question is done.

Oh wait - I get it!

We start A swinging, and A has some frequency f_a given by the usual pendulum formula. OK so far?

Bhavay

Yes.

John Rennie

The idea is that because the horizontal string is elastic it stretches as A swings, and this creates a tension in the horizontal string. We don't need to worry about exactly what form that tension takes, only that it too will oscillate at the same frequency f_a because the tension is produced as a result of A's swings.

Make sense so far?

Bhavay

No, how will it stretch ? horizontally or vertically ?

vertically means - bend like v shape?

John Rennie

It doesn't matter. We don't need to know.

Bhavay

Why does it creates tension only when A is oscillating ?

John Rennie

Jun 10, 2020 08:34

All we need to know is that the tension creates a force on the other three pendulums.

@Bhavay let me come back to this. Go with me for now.

Bhavay

Sure.

@JohnRennie In what direction?

John Rennie

So the point is that the elastic string creates a force on the other three pendulums. We don't know exactly what form that force takes and we don't care. The important thing is that the force is an oscillating force and the frequency of the oscillation is f_a, the same as pendulum A.

Bhavay

This question is raising new questions at each new point.

John Rennie

@Bhavay we don't know what direction. It doesn't matter because we can answer the question without knowing.

Bhavay

Like why is this force oscillating now?

John Rennie

Jun 10, 2020 08:38

You need to park these questions for now.

Bhavay

Parked.

John Rennie

OK. Suppose you're pushing a child on a swing. You need to push at a frquency that matches the frequency of the swing. Yes?

Bhavay

Yes , otherwise the child might fall off.

John Rennie

Well you could try pushing at a different frequency, but then the swing wouldn't swing as far because sometimes you'd be pushing when the swing was moving towards you and your push would slow it down instead of speeding it up.

You get the maximum amplitude when the applied force has the same frequency as the swing. This is the phenomenon known as resonance.

OK so far?

Bhavay

Just one question , don't we push the child only when the swing is coming back?

John Rennie

Jun 10, 2020 08:44

Typically you wait for the child to be nearest to you, then you start pushing as the child starts moving away.

Bhavay

Yes.

John Rennie

The point is that the frequency of the force you apply matches the frequency of the swing.

Bhavay

but we push the child once ,right?

and then again push ,when the swing returns to that same point.

John Rennie

Yes

Bhavay

I think i am not clear with this statement -"Force with Frequency"

John Rennie

Jun 10, 2020 08:46

The period of the swing is the interval between the times the child is nearest to you. Yes.

Bhavay

Although i am aware of resonance ex - breaking of glass when opera singer is singing, matches the frequency of glass molecule.

John Rennie

Let me draw a diagram ...

@Bhavay That's just a graph of the displacement of the swing with time. OK so far?

Bhavay

I feel stupid now, but i don't get it.

so at every trough the swing is nearest to us?

John Rennie

Maybe this is a bad example. I assumed that pushing a child on a swing was something we'd both be familiar with.

@Bhavay if the swing is at rest, i.e. hanging straight down, it is some distance away from you. Yes? You don't stand at the rest position or the swing would hit you. You stand a metre os so away from the swing.

Bhavay

Okay.Let's compare it with motion of pendulum.

I am at the other extreme?

John Rennie

Jun 10, 2020 09:01

Bhavay

That really helped.

John Rennie

Now I'll draw the force you are applying on the graph. I'll draw it in red:

Bhavay

I never really thought pushing on a swing is that much complicated.

John Rennie

So you apply a force starting when the child is nearest you and as it starts moving away. The red curve shows approximately how the force you apply varies with time.

OK so far?

Bhavay

I have a problem in this statement-

if a force is applied with a magnitude greater than frequency...Wait that cant be correct as we can't compare frequency with newtons.

Do u see where my problem is ?

John Rennie

Jun 10, 2020 09:10

When did I say anything about the magnitude of the force?

All I said was the frequency of the force has to match the frequency of the swing.

Bhavay

isn't pushing a force?

John Rennie

I didn't say anything about the magnitude.

Bhavay

okay frequency of the force means time after which u apply a force/ push the swing,right?

John Rennie

If you look at my last diagram then the force you apply (the red line) is periodic with the same period T as the swing. It's not a sine wave, but it is periodic.

Bhavay

Yes.

John Rennie

Jun 10, 2020 09:13

And the frequency of both the swing and the force is just f = 1/T

Bhavay

Yes.

John Rennie

The reason I used this example was to say that with any oscillating system, if you want to get the maximum ampitude you need to apply a periodic force with the same frequency as the oscillating object.

Bhavay

why doesn't the magnitude of force affects the amplitude?

John Rennie

Assume the magnitude of the force is fixed, and we are only varying the frequency.

Bhavay

Okay here this out- If the frequency of force is less than frequency of swing , wouldn't the amplitude increase?

Ex - If we also apply force to swing when it is at +A , won't it go higher (further)?

John Rennie

Jun 10, 2020 09:22

If the frequency of the force is less than the frequency of the swing you get behaviour like this.

Bhavay

i meant more ,sorry.

John Rennie

You find your push gets out of phase with the swing so you inevitably end up pushing when the swing is heading back towards you, and that slows the spring down.

Bhavay

Okay let's not waste any more time on this .I agree that frequency of the force has to match the frequency of the swing.

So C will oscillate with maximum amplitude?

John Rennie

Yes

Because C has the same frequency as A

That took a long time :-)

Bhavay

length*

Tbh i am still not sure how will the other pendulum start to oscillate , but let's leave that problem for another day.

John Rennie

Jun 10, 2020 09:28

Let me see if I can find a video on youtube

Bhavay

@JohnRennie Are u still free?

John Rennie

Coupled Pendulums - Sixty Symbols

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There, that shows the effect.

BJ0001 : Four coupled pendulums and finding which one has the maximum amplitude

Room for B & J

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