Problem Solving Strategies

cOnnectOrTR 12

05:09

Hello @JohnRennie

John Rennie

@cOnnectOrTR12 Hi :-)

cOnnectOrTR 12

How are you?

John Rennie

I'm good thanks. How are you?

cOnnectOrTR 12

Fine. Weather is beautiful here. :)

John Rennie

This is the best time of the year in the UK. The temperature is about 20°C so it's warm enough to be pleasant but not too hot to do anything.

I chat to a student in Rajasthan, and it's 50°C there!!!

cOnnectOrTR 12

05:14

@JohnRennie here also. Yesterday it rained and made the environment cooler after 45 degrees of heat wave.

John Rennie

45°C. That's hot! It never gets that hot in the UK.

In the part of the UK where I live the hottest it ever gets is about 30°C.

cOnnectOrTR 12

@JohnRennie yeah. You can make omelettes here on stone.

@JohnRennie That is very good. You are lucky 🍀

LearningCHelpMeV2

06:30

What does it mean for "an object to have a large enough mass for its own gravity to give it a round shape"

This was part of the definition I received for a planet

But i dont understand it

John Rennie

@LearningCHelpMeV2 Hi :-)

LearningCHelpMeV2

Good morning

John

John Rennie

Suppose you take some jagged piece of rock and you start to squeeze it. At first nothing happens because ... well ... rocks are hard! But if you squeeze it hard enough the rock will start to deform and you could squeeze it into a different shape.

OK so far?

LearningCHelpMeV2

Yup

John Rennie

Well now consider putting that rock at the centre of the Earth. The rocks at the centre of the Earth are being squeezed by the weight of all the rock above them, and that force is large enough to make the rocks flow like a liquid.

In fact the pressure rises so quickly with depth that you only need to down a few hundred miles from the surface for the pressure to get so large it can deform the rock like a liquid.

So apart from a thin shell of crust round the outside the Earth is really a large ball of liquid floating in space, and balls of liquid naturally form into spheres.

That's why the Earth is (approximately) a sphere.

OK so far?

LearningCHelpMeV2

06:43

Yup got it

John Rennie

This happens because the Earth is big enough for the rock inside it to be squeezed and deformed by the weight of the rock above it. But if you consider a smaller planet the pressure inside it is lower simply because the smaller planet contains a smaller weight of rock.

And if you make the planet small enough the pressure even right at the centre isn't large enough to make the rock flow.

In that case, since the rock can't flow it won't form into a sphere. The planet will be stuck with whatever shape it had to start with, which is probably some random jagged shape.

And that's what the statement "an object to have a large enough mass for its own gravity to give it a round shape" is referring to.

LearningCHelpMeV2

Thanks :), i need to reread this a few times, but it makes a bit more sense now

John Rennie

OK :-) I'm around for a while if you have any follow-up questions.

cOnnectOrTR 12

08:31

@JohnRennie I have a question.

John Rennie

Hi :-)

cOnnectOrTR 12

Hi 👋

John Rennie

What's the question?

cOnnectOrTR 12

I found out W and then used work energy theorem.

I WhatsApped

John Rennie

Got it, reading now ...

The hard way to do this would be to solve for the equation of motion i.e. calculate x(t)

Then you can find the time that x is at 2m, and then differentiate to find the KE at this time.

This would be a lot of work, but fortunately they have made the question easy and there's a quicker way to do it.

What have you done so far? How far have you got?

cOnnectOrTR 12

08:40

I used work energy theorem.

John Rennie

Can you show me how you did the calculation?

OK, that looks fine to me. As you say, the KE is equal to the work done, and the work done is ∫F(x)dx where you're integrating from x = 0 to x = 2.

user157860

@JohnRennie, Hi! :)

John Rennie

So where is the difficulty with finding the solution?

@user157860 Hi :-)

cOnnectOrTR 12

But maybe I can’t use that for varying force?

@JohnRennie answer is not matching.

John Rennie

Give me a moment and I'll do it ...

@cOnnectOrTR12 I get (3) 5.33 J. What answer does the book give?

cOnnectOrTR 12

08:48

2

John Rennie

8.67 J

cOnnectOrTR 12

Yes

I also got 3

John Rennie

I don't understand how they got that answer ...

cOnnectOrTR 12

@JohnRennie then it’s probably wrong. 😂

I have a stone question.

user157860

@JohnRennie, ""...1)..a wire moving through a magnetic field will have a current induced in it, but according to the contemporary understanding, there was one theory that covered the case of the magnet moving and the wire at rest, whereas 2) there was a wholly different theory for the case where the magnet was at rest and the wire moved""

just grant my quote from EB as true, does it resolve Einstein's worries?

John Rennie

08:55

@cOnnectOrTR12 That should be straightforward. The change in KE is equal to the change in gravitational PE mg𝓁.

cOnnectOrTR 12

@JohnRennie that is when conservative force acts. Here there is tension?

John Rennie

@cOnnectOrTR12 The tension acts along the string, and the length of the string doesn't change. That means the tension does no work. Yes?

cOnnectOrTR 12

@JohnRennie ok

John Rennie

@user157860 I suspect the EB article is trying to explain the history of the subject. If you go back to the late 18th/early 19th century when the laws of electromagnetism were first being discovered then what they say is true.

cOnnectOrTR 12

@JohnRennie but when we whirl a stone we are doing work. Isn’t it?

user157860

09:01

It was not, but that is not relevant: if it were true, there is no problem? and how did he resolve the proble,?

John Rennie

But in the late 19th century Maxwell unified the originally separate laws when he wrote down the equations that we call Maxwell's equations these days.

user157860

the article in EB dates 1974

John Rennie

And Einstein was well acquainted with Maxwell's laws.

user157860

ok, so how did he fix the problem with SR?

John Rennie

The trouble is I don't know what the article means by "Einstein's problem"

user157860

09:05

he was worried that result would change depending on which element was moving

John Rennie

I've read a lot about Einstein's work, and I've never come across him being puzzled in the way you describe.

cOnnectOrTR 12

Again wrong !

John Rennie

@cOnnectOrTR12 Give me a moment and I'll do the calculation.

cOnnectOrTR 12

Ok

user157860

is there no problem in relativity if we get different results if the wire or the magnet is moving?

John Rennie

09:10

@user157860 Not that I'm aware of.

@cOnnectOrTR12 I get (1) √u² - 2gL

cOnnectOrTR 12

Yes but it says 4)

user157860

Here is the full text:Einstein tells what motivated him in the opening paragraph of his 1905 paper that became known as Special Relativity:

It is known that Maxwell's electrodynamics—as usually understood at the present time—when applied to moving bodies, leads to asymmetries which do not appear to be inherent in the phenomena.

He goes on to explain that it was well known that a wire moving through a magnetic field will have a current induced in it, but according to the contemporary understanding, there was one theory that covered the case of the magnet moving and the wire at rest, whereas there was a wholly different theory for the case where the magnet was at rest and the wire moved

Take, for example, the reciprocal electrodynamic action of a magnet and a conductor. The observable phenomenon here depends only on the relative motion of the conductor and the magnet, whereas the customary view draws a sharp distinction between the two cases in which either the one or the other of these bodies is in motion. For if the magnet is in

For if the magnet is in motion and the conductor at rest, there arises in the neighbourhood of the magnet an electric field with a certain definite energy, producing a current at the places where parts of the conductor are situated. But if the magnet is stationary and the conductor in motion, no electric field arises in the neighbourhood of the magnet. In the conductor, however, we find an electromotive force, to which in itself there is no corresponding energy,

but which gives rise—assuming equality of relative motion in the two cases discussed—to electric currents of the same path and intensity as those produced by the electric forces in the former case.

This problem bothered Einstein for years. It made no sense to him that these two cases should require separate theories; he reasoned that there should only be one theory to cover both cases. SR is the solution to that problem.

cOnnectOrTR 12

@JohnRennie I don’t understand one thing. It’s the force due to which the stone is going around but is not doing work ?

John Rennie

@cOnnectOrTR12 Ah! It doesn't ask for the velocity when the string is horizontal, it asks for the change in the velocity.

user157860

apparently, these worries are in his original paper

cOnnectOrTR 12

09:16

@JohnRennie first is right even then. I guess.

It’s asking v.

John Rennie

It asks for the change in velocity

cOnnectOrTR 12

Must be Wrong printing.

user157860

@JohnRennie, can you just find me 2 links where I can learn more about the two phenomena?

cOnnectOrTR 12

@JohnRennie I don’t understand how this motion is similar to pendulum?

John Rennie

@user157860 I think what the article means is that a stationary magnetic field looks like a mixture of magnetic and electric fields if you move relative to it, which as I recall is where we came in last week.

user157860

09:23

can you suggest some links?

John Rennie

There isn't a shortcut to understanding this. You need to find a book on special relativity and read it. You can certainly Google for the equations giving the magnitude of the electric field for a moving observer, but this is just an equation that won't help you understand what is going on.

Classical electromagnetism and special relativity

The theory of special relativity plays an important role in the modern theory of classical electromagnetism. It gives formulas for how electromagnetic objects, in particular the electric and magnetic fields, are altered under a Lorentz transformation from one inertial frame of reference to another. It sheds light on the relationship between electricity and magnetism, showing that frame of reference determines if an observation follows electrostatic or magnetic laws. It motivates a compact and convenient notation for the laws of electromagnetism, namely the "manifestly covariant" tensor form. Maxwell...

That gives you the equations, but I suspect you'll find it hard going.

cOnnectOrTR 12

@JohnRennie can you tell me this is similar to motion of pendulum just with more velocity. Yes?

user157860

ok, can you spare some more minutes to conclude our previous chat on propagation?

John Rennie

@cOnnectOrTR12 It is governed by the same equations as a pendulum, but I don't think that is helpful in this case.

@user157860 Yes, though you'll have to remind me where we got to.

@cOnnectOrTR12 I don't understand how the answer can be (4) ...

cOnnectOrTR 12

@JohnRennie if there is no air then it will keep on moving in circle as the pendulum does? Isn’t it?

user157860

09:31

you said light needs no medium

cOnnectOrTR 12

@JohnRennie Mee too :)

John Rennie

@cOnnectOrTR12 Aha, got it! :-)

Let me draw a diagram ...

user157860

and Maxwell stated that light propagates in vacuum at C, and that was universally accepted even by Einstein. What escapes me is why did he/we need relativity to justify the invariance of speed? Like sound, light speed should not be affected by the velocity of emitter and receiver, right?

John Rennie

@cOnnectOrTR12 These are the two velocities. Yes?

cOnnectOrTR 12

Ok

user157860

09:36

everybody measure same speed because it is not affected by motion of the observer,

cOnnectOrTR 12

@JohnRennie Pythagoras?

John Rennie

:61203295Yes :-) The change in the velocity is final velocity - original velocity so it's this:

cOnnectOrTR 12

Got it :)

4)

John Rennie

where the blue arrow is the change.

@cOnnectOrTR12 Yes :-)

It was the terminology used in the question that was confusing me.

They wanted you to find the difference between the original and final vectors then find the magnitude of this change vector.

@cOnnectOrTR12 Are you OK with this now. Can I get on to answering @user157860 ?

cOnnectOrTR 12

@JohnRennie yes! Do the stone stops because there is air?

John Rennie

09:41

Do you mean why does it slow down if we do the experiment?

cOnnectOrTR 12

Yes

John Rennie

Yes, it's due to drag caused by the air. If you did it in a vacuum it wouldn't slow down.

@user157860 Hi, are you still there?

user157860

yep!

cOnnectOrTR 12

@JohnRennie thank you :)

John Rennie

OK. To set the stage suppose we solve the equations that describe a sound wave i.e. a wave moving through air, so here air is the medium. We find the wave moves at a velocity given by:

v = √γP/ρ

where P is the air pressure and ρ is the density of the air.

The point is that the speed is determined by the properties of the medium i.e. the parameters like pressure and density are properties of the medium.

Now we do the same for an E wave.

This is done using Maxwell's equations. The working is somewhat tedious, but again we end up with a speed in terms of some constants:

c = 1/√ε₀μ₀

where this time the constants ε₀ and μ₀ are the permittivity and permeability of free space.

@user157860 OK so far?

user157860

09:48

ok,

I asked a question, are the answers correct in your view?hsm.stackexchange.com/questions/14456/…

John Rennie

Now, nowhere in the working did we invoke a medium but it is a natural assumption that the wave must be moving in a medium, and that ε₀ and μ₀ are properties of that medium in way that P and ρ are properties of air.

And for a while physicists assumed this was the case, and they proposed a medium called the aether to explain how light could propagate through space. Then ε₀ and μ₀ are properties of this aether.

But, there's a problem and it's a big one.

user157860

but there is no aether and we ned no medium you said

John Rennie

I said: it is a natural assumption that the wave must be moving in a medium

But it turned out that the assumption was wrong.

user157860

you said it in a comment to my question at PSE

John Rennie

Proving only that making assumptions because you think they are natural is a dangerous business in physics!

user157860

09:52

ok

so light propagates in vacuum but need so medium, why should its speed be influenced by the motion of the meitter or observer?

John Rennie

The problem is that if you have a medium it's possible to have a speed relative to that medium. e.g. if I'm cycling at 10 m/s on a still day then I'm moving at 10 m/s relative to the air. Yes?

user157860

ok

John Rennie

Suppose there is a sound coming towards me through the air. The sound moves through the air at 340 m/s, but I'm also moving through the air towards the sound at 10 m/s. So the speed of the sound relative to me is 350 m/s.

user157860

ok

John Rennie

That is, I measure the speed of the sound to be 350 m/s not 340 m/s.

And this is true of any wave that moves through a medium. For example it's true of water waves as well.

user157860

09:56

but light needs no medium, right?

John Rennie

But when we calculate the speed of light using Maxwell's equations we discover something very strange - the speed of the light is the same for all observers regardless of what speed they have.

So we have a contradiction.

user157860

that should be so, because there is no medium

how can anyone influence its speed?

John Rennie

That's the point. We can only explain this if there is no medium i.e. light doesn't move through a medium.

user157860

right , that makes sense, why SR, then?

John Rennie

Because without SR we would still find the speed of the light depends on the speed of the observer, even when we don't assume a medium is present. Consider this thought experiment:

user157860

10:00

ok

John Rennie

Imagine we're in a vacuum and I throw a ball at you. The ball isn't moving through a medium, but the speed of the ball relative to you still depends on your motion.

user157860

ok

John Rennie

If you move away from me then relative to you the ball slows down, while if you move towards me then relative to you the balls speeds up.

user157860

right

but if I move toward light I just measure a higher frequency

John Rennie

Now replace the ball by a light wave. This time I flash a torch at you and again you can move towards me or away from me.

user157860

10:03

but if I move toward light I just measure a higher frequency

John Rennie

@user157860 Yes, this time you will find the light has the same speed regardless of how you were moving. All that happens is its frequency changes.

user157860

right

why SR?

John Rennie

Now Maxwell's equations tell us the speed can't change, so that's OK, but Maxwell's equations don't tell us why the speed can't change.

user157860

so what?

you said because of epsilon time mu determine it

John Rennie

So physicists being curious creatures immediately tried to understand why the speed can't change, and it turns out that it's due to a fundamental symmetry of the universe.

user157860

10:06

you said because of epsilon time mu determine it

John Rennie

@user157860 well yes, but why do ε₀ and μ₀ determine it?

You could of course say they just do but sadly we physicists are not content with that :-)

user157860

that's the property of electromagnetic field

John Rennie

That's not an explanation.

user157860

ok, go on

John Rennie

The explanation is that the universe has a symmetry called Lorentz covariance.

And special relativity is a result of this symmetry.

Now I guess you're going to say this has just replaced one assumption with another assumption, and we could ask why Lorentz covariance?

user157860

10:09

that always happens when you ask why

regressio ad infinitum

John Rennie

And I'm not sure that has an answer. All we can say is it's the simplest possible assumption because from that single assumption everything else follows.

All of SR follows from Lorentz covariance.

user157860

so, if we do not ask WHY, we can do without SR?

John Rennie

At the end of the day physicists are trying to explain the results of experiments. SR is all very pretty, but to earn its keep it has to predict the result if we do an experiment to test it.

Which fortunately it does :-)

user157860

could you add a better answer here?hsm.stackexchange.com/questions/14456/…

John Rennie

So we can't do without SR unless we give up trying to explain the world around us.

user157860

10:12

that's fair enough, at last !!!

John Rennie

@user157860 You're asking two different things there.

user157860

am I?

John Rennie

You ask how ε₀ and μ₀ were measured, and you also as why is c = 1/√ε₀μ₀

user157860

they are related

John Rennie

Remember I said how in the late 18th and 19th centuries physicists started to discover laws describing magnetic and electric fields?

Well the equations they got included the constants ε₀ and μ₀

user157860

10:15

my main worry is that you can obtain the same result just manipulating the 2 values, why are we forced to choose those values?

John Rennie

At the time they were just constants with no special significance, and they weren't though to be related because it wasn't understood how electric and magnetic fields were related.

But then Maxwell realised all these separate equations the 18th century physicists had discovered were just part of a single set of equations we call Maxwell's equations.

OK so far?

user157860

are there really 10 differential equations? If so, why did he need so many?

John Rennie

@user157860 You're thinking of Einstein's equation for general relativity. That has ten different equations.

user157860

Maxwell had 8?

John Rennie

Maxwell's equations are really a single equation, but we split it into four parts to make the calculations easier.

user157860

10:21

ok, , I suppose you find those answer to my question correct

John Rennie

Mauricio's answer looks good to me. The other answer is an extended quote that I don't think explains much.

user157860

a couple of more simple questions: 1) when we produce a radio signal we talk of amplitude which is the intensity, right? is it due to the sum of many similar waves?

or to a greater displacement of electrons?

John Rennie

If you consider a wave on water, the wave can have any height. If you disturb the water only a little you'll get a small water and if you distrurb the water a lot you'll get a large wave.

The large wave isn't the sum of smaller waves, it's just a large wave.

user157860

but can a light wave have amplitude?

John Rennie

@user157860 yes, the more you make the electrons oscillate in your aerial the greater the "height" of the EM wave.

user157860

10:25

so it is displacement the reason?

John Rennie

Though for an EM wave the "height" isn't a physical distance, it's the strength of the electric field.

user157860

I mean displacement of electrons in the antenna

John Rennie

The amplitude of the wave depends on the acceleration of the electrons in the antenna.

So it depends on both how far you displace them and how fast you displace them.

user157860

I suppose the two factors are related. 2) do photons from the sun have different amplitudes?

John Rennie

The relationship between a photon and a light wave is a complicated one ...

94

Q: What is the relation between electromagnetic wave and photon?

At the end of this nice video (https://youtu.be/XiHVe8U5PhU?t=10m27s), she says that electromagnetic wave is a chain reaction of electric and magnetic fields creating each other so the chain of wave moves forward. I wonder where the photon is in this explanation. What is the relation between ele...

A photon doesn't have an amplitude

user157860

10:30

can em waves in the visible have different amplitudes?

John Rennie

Yes, the amplitude corresponds to how bright the light is. Greater amplitude = greater brightness.

user157860

3) does a single oscillation with 1 cm wavelength have same energy of one with 1cm wavelenght?

John Rennie

I'm not sure what you're asking ...

Are you thinking a single oscillation corresponds to a photon?

user157860

consider 2 em waves one has frequancy 10^18Hz, the other 10^12 Hz, if we consider one single oscillation I suppose they have the same energy, right? one is nore energetic only because in one sec there are more oscillation, right?

John Rennie

The energy depends on the amplitude as well as the frequency.

user157860

10:40

ok John, that's all, thanks for your time :) the amplitude is a fascinating subject

John Rennie

OK :-)

Kavin Ishwaran

11:03

@JohnRennie Hi !

John Rennie

@KavinIshwaran Hi :-)

Kavin Ishwaran

If my understanding is right, Are we measuring energy of the EM waves with respect to time ?.. Suppose wave 1 and wave 2 have same wavelength and amplitude but different frequency, but each individual wave will have same energy ? .

John Rennie

In fact the energy carried by an EM wave depends only on its amplitude and not on its frequency.

e.g. suppose you are being warmed by the heat coming from a fire.

Then the frequency of the infrared coming from the fire doesn't matter. The only thing that determines how much you are being heated up is the amplitude.

Kavin Ishwaran

ohhh

Can we think like this:

I am throwing 3 balls in a second with same velocity. I have spent some energy to do this work. The total energy I have done (assuming 100% efficiency) is sum of kinetic energy of the three balls, Now I do the same with 4 balls I have done more energy but the velocity of the ball remains the same. As an anology Here each ball is an individual wave, the velocity is amplitude ?

John Rennie

Did you mean the total KE is amplitude?

Kavin Ishwaran

11:18

@JohnRennie KE of Individual ball

John Rennie

I'm not sure your analogy works. Are you thinking the balls represent the photons in the light wave?

Kavin Ishwaran

@JohnRennie yes kind of

John Rennie

The problem is that a light wave is not like a load of little balls with each ball being a photon. It's more complicated than that.

Kavin Ishwaran

Yes. But now for an anology I have compared it to balls

John Rennie

That analogy isn't going to work.

Kavin Ishwaran

11:22

ohh

John Rennie

Trying to build up a light wave from photons is very complicated because photons are particles not waves.

I must admit I don't understand how to do it.

Kavin Ishwaran

Ohh. Ok then . Thank you for sparing some time :-)

John Rennie

You're welcome. Sorry I can't help more.

Kavin Ishwaran

Its ok Sir. bye :)

Transcript for

Problem Solving Strategies