A Fistful of Brain Teasers

Mike Sullivan posted:

Get 2 cans of beer. Open and drink one. Roll them both down an incline. The full one will get to the bottom first, Open it and drink it.

Yes, but they are different weights, and inertia to overcome rolling resistance comes into it - the challenge was two identical size shape and, but different distribution of weight.

David Hendon posted:

They don’t both weigh the same though. But taking a limit case is often a good way to visualise the behaviour.

So visualise a solid cylinder mounted on a spindle with a handle. And a very large wheel with most of the weight around the rim, again mounted on a spindle with a handle. Which one would experience suggest would be the hardest to get going? Or once you had got them both going, if you stopped turning the handle, which would stop first? That is also the one that would accelerate under a constant force (gravity) fastest and so reach the bottom of an inclined plane first....

best

David

Yes, but we are not turning by hand the effort to move being an identical gravitational attraction on each. Meanwhile, whilst the force at the circumference of the lead cylinder is more as it is proportional to mass, with denser periphery than the titanium cylinder, nearer the centre the lead cylinder will have much less force applying because it is part lead and part air, so the mass of  the titanium cylinder at that same distance will be greater than the lead one so at that point there would be more force on the titanium cylinder. I haven’t done the maths, but I wonder if the net effect is the same for both.

Still nothing persuasive as to why the two should roll any differently, so I am still highly doubtful.

Innocent Bystander posted:

David Hendon posted:

They don’t both weigh the same though. But taking a limit case is often a good way to visualise the behaviour.

So visualise a solid cylinder mounted on a spindle with a handle. And a very large wheel with most of the weight around the rim, again mounted on a spindle with a handle. Which one would experience suggest would be the hardest to get going? Or once you had got them both going, if you stopped turning the handle, which would stop first? That is also the one that would accelerate under a constant force (gravity) fastest and so reach the bottom of an inclined plane first....

best

David

Yes, but we are not turning by hand the effort to move being an identical gravitational attraction on each. Meanwhile, whilst the force at the circumference of the lead cylinder is more as it is proportional to mass, with denser periphery than the titanium cylinder, nearer the centre the lead cylinder will have much less force applying because it is part lead and part air, so the mass of  the titanium cylinder at that same distance will be greater than the lead one so at that point there would be more force on the titanium cylinder. I haven’t done the maths, but I wonder if the net effect is the same for both.

Still nothing persuasive as to why the two should roll any differently, so I am still highly doubtful.

What causes them to roll down a plane? Why do you think that something where the mass is concentrated at the edge should accelerate from rest at the same rate as something where it is distributed equally throughout the cylinder?

best

David

David Hendon posted:

Innocent Bystander posted:

David Hendon posted:

 

So visualise a solid cylinder mounted on a spindle with a handle. And a very large wheel with most of the weight around the rim, again mounted on a spindle with a handle. Which one would experience suggest would be the hardest to get going? Or once you had got them both going, if you stopped turning the handle, which would stop first? That is also the one that would accelerate under a constant force (gravity) fastest and so reach the bottom of an inclined plane first....

 

Yes, but we are not turning by hand the effort to move being an identical gravitational attraction on each. Meanwhile, whilst the force at the circumference of the lead cylinder is more as it is proportional to mass, with denser periphery than the titanium cylinder, nearer the centre the lead cylinder will have much less force applying because it is part lead and part air, so the mass of  the titanium cylinder at that same distance will be greater than the lead one so at that point there would be more force on the titanium cylinder. I haven’t done the maths, but I wonder if the net effect is the same for both.

Still nothing persuasive as to why the two should roll any differently, so I am still highly doubtful.

What causes them to roll down a plane? Why do you think that something where the mass is concentrated at the edge should accelerate from rest at the same rate as something where it is distributed equally throughout the cylinder?

 

I did explain earlier on: the acceleration is only due to gravity. If in freefall they would fall at the same rate - other than if there were ridges impeding progress for which difference in inertia with the flywheel effect might make a difference, what is there to make one accelerate faster than the other?

You postulated that the different density distribution would cause one to accelerate faster than the other, citing the flywheel inertia effect where it is harder to get the wheel with a weighrted outer spinning, and then harder to stop it,  compared to a more compact weight, initiallty thinking the hollow lead one would roll taster because of being heavier at the rim, then changed to the solid titanium one.

As I said, I am not an expert, but I don’t see how your argument applies - I may be wrong, but a clear mechanical explanation is needed

Now thinking further, what makes the cylinder roll on an inclined plane is gravity on the part of the cylinder that most overhangs the centre of gravity. CoG is dead centre on a horizontal plane, so no movement. Tilt it and CoG moves slightl making one side relative to CoG heavier so it drops, which means it starts to roll - and that is a continuous effect, so it accelerates. But in what way does a heavy rim alter tgat compared to a solid weight? In the solid weight the middle section is heaviest - but still past CoG when tilted. In heavy rim, which is like a ring, the heaviest section is somewhere near the outer edge - of course balanced by the same section the other side, so it is only the bit just off-centre that is responsible for the rolling acceleration - which is lighter than that of the solid titanium cylinder. Noot sure why that should make a difference...  (That was ‘thinking out loud’, scribbling as I reasoned. I think it is totally correct, but doesn’t tell me what I need to know!)

The moments of inertia will be different.

If both cylinders are given a push to roll along the ground, the leaden one will roll farther.

Don Atkinson posted:

The moments of inertia will be different.

If both cylinders are given a push to roll along the ground, the leaden one will roll farther.

The feel if warmer or colder than body temperature stated by Fatcat is also correct (If both stable at room temp, then touching, the more conductive one, titanium, will feel colder -  though I don't know how obvious it will be as it might not be a large difference.

(Some others are also correct, but without going back over every one I think they all need something else to either do something or measure something - even the inclined plane needs that.)

As for the speed on an inclined plane, it would be nice if someone can come up with a definitive answer supported by detailed mechanical explanation...

Innocent Bystander posted:

Don Atkinson posted:

The moments of inertia will be different.

If both cylinders are given a push to roll along the ground, the leaden one will roll farther.

The feel if warmer or colder than body temperature stated by Fatcat is also correct (If both stable at room temp, then touching, the more conductive one, titanium, will feel colder -  though I don't know how obvious it will be as it might not be a large difference.

(Some others are also correct, but without going back over every one I think they all need something else to either do something or measure something - even the inclined plane needs that.)

As for the speed on an inclined plane, it would be nice if someone can come up with a definitive answer supported by detailed mechanical explanation...

David Hendon came up with the correct concept based on inertia and I think he was supported by Mike Sullivan.

I am surprised that my answer hasn't been challenged. It works. But events differ depending on whether the two cylinders are pushed with the same displacement force, or whether they are pushed so as to accelerate at the same rate before being left to decelerate.

Fatcat's suggestion, which is based on thermal conductivity might work, but the paint layer might interfere. Also the thermal conductivity of Lead is about 35 watts per metre-kelvin whilst titanium is 20 watts per metre-kelvin. ie, not a big difference. Bear in mind that wood has a thermal conductivity of about 0.133 watts per metre-kelvin. We all know that at normal room temperatures (say 21 deg C) wood "feels" warmer than metal. The warmth of our body means that heat flows from our hand into the wood or metal. The rate at which it flows determines the "feel". You can look up the tables of thermal conductivity and decide whether the lead will feel colder or the titanium.

So, let's say "well-done" to David and Mike and "jolly-good-try" to Fatcat !

Innocent Bystander posted:

Don Atkinson posted:

The moments of inertia will be different.

If both cylinders are given a push to roll along the ground, the leaden one will roll farther.

The feel if warmer or colder than body temperature stated by Fatcat is also correct (If both stable at room temp, then touching, the more conductive one, titanium, will feel colder -  though I don't know how obvious it will be as it might not be a large difference.

(Some others are also correct, but without going back over every one I think they all need something else to either do something or measure something - even the inclined plane needs that.)

As for the speed on an inclined plane, it would be nice if someone can come up with a definitive answer supported by detailed mechanical explanation...

I have had a quick look back and so far as I can tell, all the others involved damage or use of other equipment - even the "bucket" for Tobyjug's water..........which wouldn't have worked anyway !

Innocent Bystander posted:

Don Atkinson posted:

The moments of inertia will be different.

If both cylinders are given a push to roll along the ground, the leaden one will roll farther.

The feel if warmer or colder than body temperature stated by Fatcat is also correct (If both stable at room temp, then touching, the more conductive one, titanium, will feel colder -  though I don't know how obvious it will be as it might not be a large difference.

(Some others are also correct, but without going back over every one I think they all need something else to either do something or measure something - even the inclined plane needs that.)

As for the speed on an inclined plane, it would be nice if someone can come up with a definitive answer supported by detailed mechanical explanation...

That's YOUR homework for this week. I'm off to deliver a 'plane (pun intended) to Canada these next few days so might not be able to help out this time !

Innocent Bystander posted:

As for the speed on an inclined plane, it would be nice if someone can come up with a definitive answer supported by detailed mechanical explanation...

The answer is at the end of this video, but worth watching it all the way through. It's full of unexpected results.

https://www.youtube.com/watch?v=cB8GNQuyMPc

Strangely the video is titled 'Walter Lewin demonstrates moment of inertia', but inertia isn’t mentioned as the video.

I suppose he mainly teaches kids, as he says, and maybe kinetic energy Involved as the cylinder moves is a simpler concept to get across than moment of inertia.  Quite entertaining anyway. Thanks for finding the video!

best

David

Stagecoach Revisited

You’ve noticed in 'Western' films how the wheels of stage-coaches often appear to be rotating backwards.

In a film Mrs D and myself saw recently, the wheels appeared to be stationary although the horses were galloping flat out.

Fascinated by this, we counted the number of spokes and found there were twelve per wheel.

We estimated that the wheels were each three feet in diameter, based on the idea that the villain of the movie was about 6’ tall and stood about twice as tall as a wheel.

If the film was being shown at twenty-four frames per second, how fast was the stage-coach moving? (assume the wheels actually were 3 ft diameter).

I’m never confident with this type of question, but I’ll take a stab emboldened by half a bottle of red (so far). 3 foot diameter wheel has a circumference of 3pi feet. So each rotation takes the coach 3pi ft. So in the simplest case each frame has the wheel doing one total rotation, so at 24fps the coach is going 72pi ft/s. But a wheel with 12 spokes will also have spokes in the same place visually if it has gone 1/12, 2/12, 3/12...11/12 of a rotation. So assuming that the spies are visually identical it is any multiple of 6pi ft/s. (This feels  right to me, but as I suggested above I don’t find this sort of question intuitive.)

Hi Eoink, 

you have earned the other half of the bottle !

if you convert your fps into mph you should have no difficulty establishing a practical speed of the stagecoach. And I guess most of our American friends will grasp mph more easily.

well done !

Wow, I’m surprised. So 5,280 feet in a mile, and 3,600 secs in an hour, so the coach is going 6*pi/5280*3600 or roughly 4.09 * pi, multiples of 12.8mph to 3sf.

Eoink posted:

Wow, I’m surprised. So 5,280 feet in a mile, and 3,600 secs in an hour, so the coach is going 6*pi/5280*3600 or roughly 4.09 * pi, multiples of 12.8mph to 3sf.

Yup ! so 12.8 mph or 25.6 mph would seem possible speeds to me. 38.4 mph seems too fast, even for Wells Fargo and John Wayne

I tackled it slightly differently.....

Let n be any integer, (representing the number of spokes rotated per frame)

in 1/24th sec the wheel rotates n/12 revs = 120*n rpm = 120 * 60 * n revs per hour    eq (1)

1 rev = 2πr = 3π feet.  eq (2)

Eq (1) and (2) → Speed = 120 * 60 * n * 3 * π  feet per hour → (120 * 60 * n * 3 * π) ÷ 5,280 mph

Sub for π = 22/7

Speed = n * (90/7) = 12.85n mph ie 12.85 when n=1 and 25.75 when n=2

Got time on your hands ...............

The hour hand and the minute hand on a clock (or analogue watch) travel at different speeds.

There are certain times when the hands are exactly opposite each other. This occurs eleven times every twelve hours.

Can you give a simple formula for calculating when these times occur?

We are dealing with a conventional 12 hour analogue clock/watch.

(2n-1)*360/11 in minutes with n= 1,2,3,....11 

Nice formula Ian !

Mine is slightly different. I'll post it later.

Position of the hour hand (in minutes past 12:00) is given by :-

(2n-1)*30/11 (with n representing the hours = 1,2,3 etc)

Position of the corresponding minute hand is (obviously) 30 minutes ahead, (or behind as appropriate)

eg just before 4:00, (n=4) with the hour hand at 19¹/₁₁, the minute hand will be at 49¹/₁₁

and at 6:00 (n=6) with the hour hand at 30 minutes, the minute hand will be at 60 minutes (or 12:00)

Hi Ian,

First musings (i'm doing this on a scrap of paper as I sup a glass of after-diner wine watching the Mrs Wilson), suggest around about 11 minutes past midnight.

This is based on the Cosine Rule in an SAS triangle, which suggests an angle of about 60 degrees between the two hands is when the greatest rate of change occurs in the third side.

At 11 minutes past the hour, the angle between the two hands will  be about 60¹/₂ degrees.

But I could be way off mark. I'll have a more careful look with calculator tomorrow , unless somebody else beats me to it !

Don Atkinson posted:

Hi Ian,

First musings (i'm doing this on a scrap of paper as I sup a glass of after-diner wine watching the Mrs Wilson), suggest around about 11 minutes past midnight.

This is based on the Cosine Rule in an SAS triangle, which suggests an angle of about 60 degrees between the two hands is when the greatest rate of change occurs in the third side.

At 11 minutes past the hour, the angle between the two hands will  be about 60¹/₂ degrees.

But I could be way off mark. I'll have a more careful look with calculator tomorrow , unless somebody else beats me to it !

I think armed with a calculator instead a glass of wine you'll be at the exact answer PDQ by the looks of it :-)

Slightly more refined at 10mins : 54 secs past midnight.

I'll try to post details later. But it's not an elegant, pure maths style approach. More like a sledge-hammer !