Gas Laws - pumped tires exploding at the top of a mountain

[i_love_science]

When you pump a bicycle tire before riding up a mountain, the tires may explode. I think it is because of Boyle's law, where since the external pressure decreases, the volume of the tire is supposed to increase and it explodes. The solution says that as the air in the tire expands on heating, due to friction with the road surface, the internal pressure increases.

Am I correct, and if not, where did I go wrong? Thanks.

 

[DaveC426913]

Those two causes are not mutually exclusive. They're both contributing factors, it's more a question of the solution suggesting one is more relevant.

Actually, the solution as you have written it cannot be correct, since it mentions nothing about altitude or lower pressure.

IOW: "as the air in the tire expands on heating, due to friction with the road surface, the internal pressure increases" is every bit as true at sea level as it is at altitude.

So, what is the exact wording of the solution?

 

[DaveE]

Yep another poorly worded homework question. The question asks about going up in elevation, but then they give an answer that includes heating on a road surface that wasn't mention in the question? What if you put the bike on the top of your car and drive up there? What if you drive up there and puncture the tire with a nail? What if a meteor hits the tire? Put too close to a campfire at the mountain top?

OK, I'll stop now... I think I've made my point. You could be correct. Where you went wrong was in having an instructor that writes confusing question/answers.

The tire may burst because the material is stretched too much. Too much tensile force for the material's strength. It's easier to think of if you consider a thin balloon instead of a tire, I think. Since PV=NkT (I know, everyone else learned it as PV=nRT, they're probably more right than me, LOL), V can increase due to the temperature rising, the pressure on the gas decreasing (the ambient pressure plus the pressure from stretching the elastic material), or from someone putting more stuff inside. So, it probably was stretched too much as the volume increased from one of those things.

Or it could have been hit by a meteor.

 

[jbriggs444]

When you pump a bicycle tire before riding up a mountain, the tires may explode. I think it is because of Boyle's law, where since the external pressure decreases, the volume of the tire is supposed to increase and it explodes. The solution says that as the air in the tire expands on heating, due to friction with the road surface, the internal pressure increases.

Am I correct, and if not, where did I go wrong? Thanks.

We are left to guess at the question since it is not shown.

A typical tire is approximately rigid. It does not expand significantly when the external pressure is removed. However, when the external pressure is removed, the material in the body of the tire (the reinforcing cords embedded in the rubber) must balance all of the internal pressure. External air pressure no longer assists the tire in this responsibility.

At the risk of over-simplification, if you look at a small patch on the tire, it is subject to three forces: internal pressure, external pressure and stress from the rest of the tire. Newton's second law applies:$$\sum F = ma$$Since we have identified the three forces, we can rewrite this as:$$P_\text{int} - P_\text{ext} - S = ma$$[Here I adopt a sign convention where outward is positive. The tire's internal pressure acts outward. The external pressure and the stress on the tire both act inward].

For a tiny bit of tire, $ma$ is negligible. In any case, it does not depend on altitude. We can treat it as zero and solve for $S$, the stress on the tire:$$S = P_\text{int} - P_\text{ext}$$It should be obvious that if $P_\text{ext}$ is reduced to zero, $S$ increases by a corresponding amount. If the stress S exceeds the breaking strength of the tire material, the tire pops.

Mind you, this tire must have been flimsy or badly over-pumped. The things have normally have safety margins that are more than adequate to deal with this sort of thing.

Bicycle tires do not heat significantly while in use.

 

[Frodo]

The stress in the tyres is due to the difference between the internal pressure trying to burst the tyre and the external atmospheric air pressure supporting the tyres.

Now, the difference in atmospheric air pressure between sea level and the top of any mountain you could ride up will only be a few pounds per square inch.

For example, the atmospheric pressure at the top of the 29,000 ft Mount Everest is about one third of that at sea level - say about 5 lb/square inch (psi), compared with 14 lb/square inch at sea level.

So, the tyres will experience an apparent increase of pressure of 14 - 5 = 9 lb/square inch.

As bicycle tyres are typicality pressurised to 70 or 100 lb/square inch (psi) it isn't going to make much difference. The increased stress is well within the safety zone for over pressurising a bicycle tyre above its recommended pressure.

More realistically, the atmospheric air pressure at the top of a 10,000 ft mountain is about 70% of sea level - say 10 lb/square inch, so the change is only 4 lb/square inch.

See How to Achieve the Perfect Bike Tire Pressure

[Bicycle] Road tires typically require 80 to 130 psi (pounds per square inch); mountain tires, 25 to 35 psi; and hybrid tires, 40 to 70 psi.

It applies to car tyres too - how often do you see reports of motorists being stranded on mountains because their tyres have burst? Or Tour de France cyclists having their tyres explode on mountain finishes?

As a child I lived at an altitude of about 5,000 ft. Mercury barometers typically stood at 66cm, not 76cm; and water boiled at 95C.

 

[i_love_science]

Here's the question:
A road cyclist pumps his tires up very hard before a trip over a high mountain pass at high altitude. Near the summit one his tires explodes. Suggest why this may have occurred.

Thanks.

 

[DaveC426913]

Okay, and what did the solution say?

 

[Frodo]

Here's the question:
A road cyclist pumps his tires up very hard before a trip over a high mountain pass at high altitude. Near the summit one his tires explodes. Suggest why this may have occurred.

You have all the information you require to answer this question in the thread above.

Your assumption in the Subject that it has something per se to do with the gas laws is leading you astray as it is incorrect: neither the pressure, volume nor temperature of the tyre gas vary (to a first order). Other things dominate.

 

[jbriggs444]

Here's the question:
A road cyclist pumps his tires up very hard before a trip over a high mountain pass at high altitude. Near the summit one his tires explodes. Suggest why this may have occurred.

Thanks.

Temperature could have made the rubber (or other carcass fibers) brittle, promoting failure.

The less traveled roadways near the mountain peak could have had sharper gravel causing a laceration and sudden failure.

Time and fatigue could have finally caused the tire to fail with no relationship to altitude at all.

A band of Afghan guerrilla fighters near the mountain peak could have launched an RPG at the bicycle, catching it in the tire.

It is an interesting exercise to compare the adiabatic lapse rate (about 6.5 K per 1 km altitude) with the change in atmospheric pressure with altitude (about 10% per 1 km altitude). If the ambient temperature is around 300 K then the 6.5 K per 1 km means a decline of a bit over 2% per km.

This means that if the bicycle tires are inflated to 5 atmospheres or greater the decline in internal pressure (2 percent of 5 atmospheres per km) will more than match the decline in external pressure (10 percent per km). The decline in external pressure cannot then explain the tire failure if the tires were originally pumped up to 75 PSI 60 PSIG or more. Must have been those pesky Afghan guerillas.

 

[DaveC426913]

So what are we concluding here, that this is a trick question? That the gas laws are not relevant (because the factors are too small)?

 

[jbriggs444]

So what are we concluding here, that this is a trick question? That the gas laws are not relevant (because the factors are too small)?

My guess is that the student is being asked to explain the predicted result of an experiment that was not run. The problem setter clearly has a particular explanation in mind. But upon closer examination, that explanation does not actually work.

The experiment, had it yielded the expected result, would have been a false positive. A better experiment with a more carefully specified set of controls could have yielded a true positive.

 

[Frodo]

The barometer question is an example of an incorrectly designed examination question demonstrating functional fixedness that causes a moral dilemma for the examiner.

Question: Show how it is possible to determine the height of a tall building with the aid of a barometer

Expected answer: Calculating the difference in pressure at the top and bottom of the building

Equally valid answers:

• Tying a piece of string to the barometer, lowering the barometer from the roof to the ground, and measuring the length of the string and barometer.
• Dropping the barometer off the roof, measuring the time it takes to hit the ground, and calculating the building's height assuming constant acceleration under gravity.
• When the sun is shining, standing the barometer up, measuring the height of the barometer and the lengths of the shadows of both barometer and building, and finding the building's height using similar triangles.
• Tying a piece of string to the barometer, and swinging it like a pendulum both on the ground and on the roof, and from the known pendulum length and swing period, calculate the gravitational field for the two cases. Use Newton's law of gravitation to calculate the radial altitude of both the ground and the roof. The difference will be the height of the building.
• Tying a piece of string to the barometer, which is as long as the height of the building, and swinging it like a pendulum, and from the swing period, calculate the pendulum length.
• Marking off the number of barometer lengths vertically along the emergency staircase, and multiplying this with the length of the barometer.
• Trading the barometer for the correct information with the building's janitor or superintendent.

 

[Frodo]

So what are we concluding here, that this is a trick question? That the gas laws are not relevant (because the factors are too small)?

I think you must have missed my " ... (to a first order). Other things dominate." qualification. I did not say irrelevant.

I was also assuming, as it wasn't stated, it was an early high school leve question and not an undergraduate level question where I would expect a more complete explanation.

 

[jbriggs444]

The barometer question is an example of an incorrectly designed examination question demonstrating functional fixedness that causes a moral dilemma for the examiner.

The Crooke's radiometer is an apparatus for which incorrect explanations were offered prior to the correct one.

 

[DaveC426913]

I did not say irrelevant.

Nor did I say you did.

I was casting the question wide. Are we all zeroing on the idea that this question is not really about atmospheric pressure, but other factors?

 

[hutchphd]

This evolved to an ad hominem attack on the creator of the question, probably well-deserved because of the lack of much needed specificity. As Feynman repeatedly pointed out elementary Physics needs to be correct

 

[Frodo]

There is an identical problem when a diver ascends from depth, from a place of high external pressure to a place of lower external pressure. She must exhale to remove air from her lungs or she will damage them.

It is important not to hold the breath, to avoid over-expansion of the air in the lungs due to pressure decrease as the depth decreases, which could cause the lung tissues to tear.

 

[sysprog]

I like @DaveE's conjectural possibility answer at the end of post #3:

Or it could have been hit by a meteor.​

 

[Frodo]

I like @DaveE's conjectural possibility answer at the end of post #3:

Or it could have been hit by a meteor.​

Me too. But don't forget the increased cosmic ray density at that altitude; nor the increased UV intensity - UV rots rubber; nor quantum mechanical effects as all those tyre gas molecules may move in the same direction at the same time and force the tyre off the rim; nor altitude sickness as it could have made the cyclist become disoriented and puncture the tyres.

 

[hmmm27]

The cost of a box of thumbtacks is less than the cost of recalling and reprinting thousands of textbooks over a typo.

The real reason (which work in conjunction with the two obvious ones) is tracks aren't well maintained. Which has a physics component, as well.