Maths conundrum.

[ollypenrice]

A friend sent me a link to this intriguing video.

I watched the first part and found myself amazed by what it revealed, but I didn't see what was coming once it was applied to astronomy. (I'm not a mathematician!)  However, I've studied the sideareal and solar day on an astronomy course and performed observations to measure the sidereal day, so I really should have spotted the connection. D'oh.

Olly

[tomato]

An intriguing  video indeed, thanks for posting. It's all about the observer's position (again!).

[iantaylor2uk]

I think the best way to think about it is that the centre of circle A is a distance 4r from the centre of circle B (where r is the radius of circle A) .

[Mandy D]

Before I jump in and state the obvious answer, what is the exact wording of the actual question on the exam paper? I'm assuming it is not what is printed on the picture presented, here! No, i cannot go and look at the video at present.

[saac]

1 hour ago, Mandy D said:

Before I jump in and state the obvious answer, what is the exact wording of the actual question on the exam paper? I'm assuming it is not what is printed on the picture presented, here! No, i cannot go and look at the video at present.

 

[Mr Spock]

How could they get that wrong? I said four as soon as I saw the question and before the answers came on screen. And I haven't done that kind of maths for over 50 years...

[Mandy D]

This is related to epicyclic gearing problems. It also applies to motor-vehicle wheels, but in a slightly different way that means the top of the tyre is travelling forwards at twice the vehicle's forward speed, whilst the bottom of the tyre is not moving forwards at all. Of course, it is all resolved by accelrations.

A good few years back, when Thrust SSC broke the sound barrier, the engineering team noted that more drag was being exerted on the car than expected before it went supersonic or even trans-sonic. My immediate reaction was that the tops of the wheels were already moving at supersonic speeds whilst the car was only travelling at half the speed of sound, hence creating supersonic drag. What they found the cause to be, I don't know.

[ollypenrice]

On 09/12/2023 at 20:03, Mandy D said:

This is related to epicyclic gearing problems. It also applies to motor-vehicle wheels, but in a slightly different way that means the top of the tyre is travelling forwards at twice the vehicle's forward speed, whilst the bottom of the tyre is not moving forwards at all. 

 

This is an interesting point. Not being a mathematician, I look at it conceptually. It seems to me that 'the bottom of the tyre' is an elusive concept because it is not defined by any property of the tyre itself (such as a mark) but by the observer who notes that every part of the moving tyre is, at some point, the bottom of the tyre. 'The bottom of the tyre' is defined by its position relative to the road. An observer looking at a car moving east to west will define 'the bottom of the tyre' as the bit touching the road and will also note that that the bit touching the road does move, east to west, which is contrary to your statement that it is not moving at all. For the roadside observer the bottom of the tyre is a point of contact which certainly is moving. 

That seems to me to be the easy bit. The difficult bit is working out exactly what point of observation discovers no movement forward at the bottom of the tyre. I suppose a number of rubber molecules will be able to shout out, very briefly, 'We are now pinned to the road which we know is not moving!'

lly 

[Mandy D]

10 minutes ago, ollypenrice said:

This is an interesting point. Not being a mathematician, I look at it conceptually. It seems to me that 'the bottom of the tyre' is an elusive concept because it is not defined by any property of the tyre itself (such as a mark) but by the observer who notes that every part of the moving tyre is, at some point, the bottom of the tyre. 'The bottom of the tyre' is defined by its position relative to the road. An observer looking at a car moving east to west will define 'the bottom of the tyre' as the bit touching the road and will also note that that the bit touching the road does move, east to west, which is contrary to your statement that it is not moving at all. For the roadside observer the bottom of the tyre is a point of contact which certainly is moving. 

That seems to me to be the easy bit. The difficult bit is working out exactly what point of observation discovers no movement forward at the bottom of the tyre. I suppose a number of rubber molecules will be able to shout out, very briefly, 'We are now pinned to the road which we know is not moving!'

lly 

Look at a track laying vehcile, what you would likely call a "caterpillar tractor" and you will soon see that the part in contact with the ground is certainly not moving forwards and extrapolate from there. You will find that the top of the track is moving, relative to the ground, at twice the forward speed of the vehicle. Is that clearer for you? Remember, all rotational motion involves constant acceleration, even at uniform angular velocities!

Edited December 12, 2023 by Mandy D

[Zermelo]

The point of the wheel in contact with the ground is instantaneously at rest, with respect to both the road and a stationary external observer. Over time, from that perspective, the point on the wheel circumference traces out a cycloid, with speed varying from 0 to twice the vehicle speed. From the perspective of the wheel hub, the point on the wheel edge traces out a circle at constant speed, and is never at rest.

[Mandy D]

@Zermelo It can't be put much better than that! Cycloid, that is the word I was trying to think of. Thank you.

[Mr Spock]

There's your cycloid 

[ollypenrice]

17 hours ago, Mandy D said:

Look at a track laying vehcile, what you would likely call a "caterpillar tractor" and you will soon see that the part in contact with the ground is certainly not moving forwards and extrapolate from there. You will find that the top of the track is moving, relative to the ground, at twice the forward speed of the vehicle. Is that clearer for you? Remember, all rotational motion involves constant acceleration, even at uniform angular velocities!

I understand this point completely but am saying that there is an alternative description which I think is also valid.

1) We can define the bottom of the wheel as being the strip of rubber molecules in contact with the ground. In this description there is no relative movement between these molecules and the ground at the instant of contact.

2) We can define the bottom of the wheel as the point of contact between the tyre and the road. In this definition, which is also valid, no specific rubber molecules appear in the description. Indeed, we can ignore them entirely  and define the point of contact as lying perpendicularly below the centre line of the axle. This point is not stationary relative to the road (unless you momentarily stop time., in which case everything is stationary.) It is moving constantly along the road.

In the first definition we are looking at relative motion between a strip of molecules and the road and in the second we are looking at movement between a geometric point and the road. Consider these definitions as arising from the observer's point of view.

- An observer on the tyre will, at the top of their rotation, see themselves as moving over the road at twice vehicle speed and at the bottom, while being squashed, as not moving relative to the road at all.

- An observer at the roadside is not going round with the tyre and observes the point of contact as moving at vehicle speed continuously.

When they argue about this in the pub afterwards, the tyre rider will exclaim, 'While I was being squashed I was stuck fast onto the road and not moving at all, relative to it. And I was certainly at the bottom of the wheel.' The roadside observer will counter by booming, 'You were at the bottom of the wheel at that instant but most of the time you weren't. This isn't just about you! It's about the bottom of the wheel. I was pointing a laser at the bottom of the wheel as the car drove along and the laser never stopped moving.'

In my opinion they are both right, the ambiguity arising from what we consider to be the bottom of the wheel. 

Olly

 

[Mandy D]

@ollypenrice I'm simply not going to discus this topic any further with you, as it would appear to be a pointless excersize. I've laid out what is going on and others have taken the time to explain further. There is absolutely no ambiguity regarding what is the bottom (or any other point) of the wheel, with regard to a discussion of the behavoir in the limit.

[ollypenrice]

33 minutes ago, Mandy D said:

@ollypenrice I'm simply not going to discus this topic any further with you, as it would appear to be a pointless excersize. I've laid out what is going on and others have taken the time to explain further. There is absolutely no ambiguity regarding what is the bottom (or any other point) of the wheel, with regard to a discussion of the behavoir in the limit.

Ouch, sorry, I seem to have offended you. That absolutely wasn't my intention and I was finding the discussion amicable and interesting. 

I believe that their is a semantic ambiguity in the term 'the bottom of the wheel.'  The point of contact does move relative to the road because it is not a point attached to any one point on the tyre.. The rubber of the tyre in contact does not. If the point of contact did not move the car would not move.  We may have, here, an example of the difference between verbal and mathematical descriptions.

Anyway, I repeat my apologies and my assurance that no offence was intended.

Olly

[wesdon1]

I haven't watched the vid yet, but based on my own experiences, I'm almost certain the tyres tread is actually accelerating then decelerating rapidly during each full turn/circle relative to the road and vehicle. Think about it...imagine a white marker at the 11:45 clock position on the outer edge of the tyre, now imagine it in roatation...it will speed up relative to the vehicle/road as it passes 11:45, then slow down as it goes past the 12:15 position, then approaching 12:45 it will start to speed up again ( again, all relatively speaking ) and so on and so forth. I've actually often wondered, how much effects does the earths orbit around the sun get affected by whether it's heading towards and away the core of our gaslaxy, during it's full yearly orbit? because the exact same principles apply here with the tyres rotation/velocity relative to the vehicle and road. You are sometimes moving towards the core, and sometimes moving away, which in Newtonian classical mechanics should have an affect, albeit small, on the relative velocities...??

That's my take folks! 

Edited December 13, 2023 by wesdon1

[Zermelo]

I think I see what you're getting at, Olly.

1 hour ago, ollypenrice said:

a semantic ambiguity in the term 'the bottom of the wheel.'

Agreed, you could take this to mean either:
(1) A specific, physical point on the wheel's outer edge that is, at the moment under consideration, in contact with road, or
(2) The location in space identified by the intersection between the wheel's outer edge and a line drawn directly downwards from the wheel's centre

They happen to be co-incident for the purpose of the current discussion.

I suspect an information modeller would represent the physical points on the wheel circumference as one concept, uniquely identified by (for example) the angular offset θ from some reference zero on the wheel, while the concept (2) above would be more like a role ("bottom of wheel") that can be fulfilled by any of the circumferential points at some appropriate time t (another role of interest might be "top of wheel"). These roles are defined relative to the vehicle, not the wheel. If the motion of the car were known, it could be modelled as a constraint between θ and t (quite simply, if the vehicle's velocity is constant).

 

2 hours ago, ollypenrice said:

The point of contact does move relative to the road because it is not a point attached to any one point on the tyre

I might choose different words, but yes, in sense 2 above, the "point" of contact does move relative to the road, even though any "point" on the wheel (sense 1) that is fulfilling the role of "bottom of wheel" at a given instant will be at rest, relative to the road.

 

[ollypenrice]

5 minutes ago, Zermelo said:

I think I see what you're getting at, Olly.

Agreed, you could take this to mean either:
(1) A specific, physical point on the wheel's outer edge that is, at the moment under consideration, in contact with road, or
(2) The location in space identified by the intersection between the wheel's outer edge and a line drawn directly downwards from the wheel's centre

They happen to be co-incident for the purpose of the current discussion.

I suspect an information modeller would represent the physical points on the wheel circumference as one concept, uniquely identified by (for example) the angular offset θ from some reference zero on the wheel, while the concept (2) above would be more like a role ("bottom of wheel") that can be fulfilled by any of the circumferential points at some appropriate time t (another role of interest might be "top of wheel"). These roles are defined relative to the vehicle, not the wheel. If the motion of the car were known, it could be modelled as a constraint between θ and t (quite simply, if the vehicle's velocity is constant).

 

I might choose different words, but yes, in sense 2 above, the "point" of contact does move relative to the road, even though any "point" on the wheel (sense 1) that is fulfilling the role of "bottom of wheel" at a given instant will be at rest, relative to the road.

 

I'm relieved since I couldn't see anything wrong with my argument. You've expressed the two interpretations of the phrase better than I did and located them both in a mathematical framework. Most kind!

Olly

[Zermelo]

2 hours ago, wesdon1 said:

the tyres tread is actually accelerating then decelerating rapidly during each full turn/circle relative to the road and vehicle

As @Mandy D said above, the tread will be accelerating even with respect to the wheel hub or vehicle, as circular motion is accelerated.  When you consider it from the perspective of the road or external observer, you add in a constant horizontal velocity (assuming the vehicle speed is constant) to that circular, accelerated motion, resulting in the cycloid. I think the instantaneous direction of the acceleration is still towards the centre of the wheel at all times (the derivative of a constant velocity offset will always be zero and won't contribute), so "accelerating" vs "decelerating" would need to be defined relative to a specific direction in the road's frame of reference.

[wesdon1]

On 13/12/2023 at 16:42, Zermelo said:

As @Mandy D said above, the tread will be accelerating even with respect to the wheel hub or vehicle, as circular motion is accelerated.  When you consider it from the perspective of the road or external observer, you add in a constant horizontal velocity (assuming the vehicle speed is constant) to that circular, accelerated motion, resulting in the cycloid. I think the instantaneous direction of the acceleration is still towards the centre of the wheel at all times (the derivative of a constant velocity offset will always be zero and won't contribute), so "accelerating" vs "decelerating" would need to be defined relative to a specific direction in the road's frame of reference.

@Zermelo Yes sorry, I should clarify that I meant accelerating and decelerating in the horizontal axis, relative to the car and road. Thank You for pointing out my unintended ambiguity!

Regards, Wes