A Fistful of Brain Teasers

Cyclist is travelling at 12/60 = 0.2 miles/minute. It will take 6/0.2 = 30 minutes to reach the bridge. The cyclist running 25 minutes late, the train takes 5 minutes to reach the bridge, traveling 6 miles. Train is travelling at 6/5 = 1.2 miles/minute or 72 miles per hour.

But if it’s British Rail, there was a leaf on the line and the train was late by 4 hours.

Yes, I know ! this one is old as the hills. But at least HH won't have to resort to the internet or tap up his mathematical genius of a son in order to get the answer

Three men order a bottle of wine at £30 to drink with their meal. They each contribute £10 for it.

The waitress asks the manager, who only has a bottle at £25, so he gives the waitress this wine and £5 change. She returns to the table but gives only £3 back, keeping £2 for herself.

Since each man paid £10 and received £1 change he has paid £9. But 3 x 9 = £27, which with the £2 kept by the waitress makes £29.

Where did the pound go? .........(no Brexit jokes please !)

Oh ! go on then. Post as many Brexit jokes as you like !

And for a bit of variety.......and just as easy !

I have two cylinders. The first one made of lead and the other of titanium.

They are identical in physical dimensions and is each painted red, so that you cannot tell which is which.

They both weigh the same, the lead cylinder being hollow and the titanium solid.

The hollow cylinder, being lead, does not sound hollow.

How can you distinguish between the two cylinders without scratching or damaging either and without using any other object ?

fatcat posted:

Place them both somewhere they will get hot. When you pick them up, the lead will feel hotter, due to its higher coefficient of thermal conductivity.

Alternativley, if they where cooled down, the lead would feel colder.

 

Or, if you have suitable measuring capability, lead has a higher thermal expansion: a 30cm lead cylinder would increase in length by nearly half a millimetre more than a titanium one if plunged into boiling water to heat from room temperature (this assuming the paint is not damaged by the boiling water). (Yes I had to look that up!)

But much quicker and easier than the above is test with a magnet, as unlike lead, titanium is weakly paramagnetic, and should be attracted to a magnet. It may be easiest to test by bringing  hanging a small strong magnet on a thread and bringing a vertical wall of each cylinder in turn close to the magnet. But that is using another object, so doesn't count. So of course is the boiling water to heat, so Fatcat's answer may be best - they only have to be a few degrees above or below body temp.

David Hendon posted:

Roll them down an inclined plane and the lead one will get to the bottom first.....

best

David

Only if you let go of it first!  Anyway, Don might complain if he is flying the plane...

However I did stop to wonder about this - would the outer ring of denser material in the lead cylinder increase acceleration in a gravity-induced rolling situation? In free-fall it makes no difference. And the rolling resistance is the same with the same surface paint and shape, size and mass of the objects. On that basis I think not

The rolling resistance is the same, but I am suggesting that the acceleration would be different between the two cylinders as they rolled down the plain (not plane!) because the weight is concentrated further from the surface of the plain in one case. There is no rotation in free fall, so that is a different case.

best

David

Plane was fine, (as opposed to ‘plane). 

I’m just not sure about the acceleration. Why, given that the only driving force is gravity: neither weight nor density affect the acceleration due to gravity, with speed in freefall the same if wind resistance the same, so if the outer part of the cylnder is heavier or lighter shouldn’t make a difference, unless there’s some factor of which I’m unaware?

 

Because the turning moment for the two cylinders is different.....

Or perhaps since they start stationary it’s about inertia, in which case I think, contrary to what I said before, that the solid cylinder will get to the bottom first because gravity acting against its inertia will accelerate the solid cylinder more quickly.

best

David

I preface this by clarifying that I am not an expert in mechanics,so could be missing something, but turning moment is force: why would that make it accelerate faster?

That make me think  - it would get over bumps better, so on an inclined plane with transverse ridges  the lead cylinder may roll faster, so maybe that is an answer?

Now, that is running into inertia - but the reverse of your alternative: and I still am inclined to think that on an a smooth surface the two would roll the same.

Why wouldn’t it make a difference? The cylinders are stationary at the top of the plane and we all agree that left to themselves they will end up at the bottom, except that if they weighed nothing presumably they would stay where they were placed.

This propensity to roll has to be due to gravity acting on them. Gravity is a constant and it acts everywhere equally. So if the mass of the cylinder is distributed differently, as it is in this case, then the acceleration caused by the gravity would be different and one will reach the bottom before the other. I now think that is the solid one.

best

David

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

TOBYJUG posted:

The lead one will still float on water ?

Not sure whether this was intended to be a serious question or not, but it will only float it is displaces enough water to compensate for the weight of the lead that is there, which given that lead is much more dense than water seems unlikely.

best

David

hungryhalibut posted:

Could you simply try bending them, or does that count as damaging them? It shouldn’t do, as because lead is bendy and titanium isn’t, the lead one could be straightened out again. 

If you bend it, then you have stretched the metal on one side and it’s very hard to unstretch bent metal, cf if you have ever tried straightening out a dent in a car. So I think that is damaging it....

TOBYJUG posted:

Lead is very heavy.  Given a certain size the ratio of density versus the amount of air inside could could cancel out and leave it floating or sinking slowly rather than the aluminium one dropping like a lead balloon.

No this is completely wrong.  They are identical in size and they both weigh the same, so the volume of water they displace will be the same. The fact that one is hollow is unknowable from an immerse-in-water point of view!

best

David

Mike Sullivan posted:

The speed of rotation is dependent on the distance of the mass from the centre of rotation.

The hollow cylinder has a higher mass away from its centre. If you spin both cylinders, the flywheel effect should keep the lead cylinder spinning longer.

David Hendon posted:
TOBYJUG posted:

The lead one will still float on water ?

Not sure whether this was intended to be a serious question or not, but it will only float it is displaces enough water to compensate for the weight of the lead that is there, which given that lead is much more dense than water seems unlikely.

best

David

It won’t - the overall density of the entire cylinder including the internal void is the same as one of solid titanium, so densrvthan water.

Mulberry posted:
Mike Sullivan posted:

The speed of rotation is dependent on the distance of the mass from the centre of rotation.

The hollow cylinder has a higher mass away from its centre. If you spin both cylinders, the flywheel effect should keep the lead cylinder spinning longer.

Good point - but not sure that a cylinder will be easipy to start spinning on one end?

David Hendon posted:
hungryhalibut posted:

Could you simply try bending them, or does that count as damaging them? It shouldn’t do, as because lead is bendy and titanium isn’t, the lead one could be straightened out again. 

If you bend it, then you have stretched the metal on one side and it’s very hard to unstretch bent metal, cf if you have ever tried straightening out a dent in a car. So I think that is damaging it....

And we have no idea of size - it might be just a few mm in diameter, many cm - the latter would not be easy to bend!

TOBYJUG posted:

Chuck them both on a bonfire and see which one melts first..

Assuming you have paint capable of  withstanding at least lead’s 328ºC melting point, you have a 50-50 chance of success: put just one in the fire (pizza oven etc), just above that temp, and wait: if it doesn’t melt and comes out undamaged you know the answer and have not failed - but if the lead one goes in and melts it will be damaged so you’ll fail!

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