While I was on my trip, I carried a water bottle with me everywhere to survive the heat. It had the typical screw top, and I wondered whether squeezing the bottle could cause the top to unscrew, or if the friction would be too great.
There are two forces acting on the cap when you squeeze the bottle: the increased air pressure in the bottle, and the normal force from the screw threads. We can diagram them like this:
where
A is the area of the bottle cap,
p is the pressure increase in the bottle,
N is the normal force applied by the threads, and
θ is the angle the threads are tilted at. The magnitude of
N is equal to the force applied directly against the threads, so we can write
Using this, we can find the torque on the cap applied by friction.
where
r is the radius of the cap,
μ is the coefficient of friction, and we have assumed that
A is given by the usual area of a circle.
In addition to the frictional torque, the tilt of the normal force will also apply a torque, given by
Combining these into a net torque gives, with some rearranging,
This says that any amount of pressure will cause the cap to unscrew, as long as
Plugging in a typical coefficient of friction for plastic, 0.2, gives an angle of about 11.5°. Looking at the water bottle I was using,
we can see that although my bottle would not work, one with only slightly steeper threads could. I think it's interesting that the pressure doesn't factor in – as long as the angle is correct, any amount of pressure can make the cap unscrew.
I can never remember how this works with friction. It seems like it takes more force to start moving an object than it does to keep moving it. Is that true?
ReplyDeleteYes, that's the distinction between static and kinetic friction. There are some materials that have interesting properties in this respect – I think certain metals will actually bind together giving an enormous static friction, but once you get them moving, the kinetic friction is much smaller.
ReplyDeleteUm, I find the "any amount of pressure" part utterly counterintuitive. For instance, when you take the bottle upstairs, the pressure inside it stays the same but the external air pressure drops by a tiny amount. Do you really say that would make the top unscrew?
ReplyDeleteSorry, I don't have time right now to work through the math myself, or I'd make a more algebraic critique...
According to my model, yes, it would start to unscrew, but probably imperceptibly slowly. Also, a real water bottle probably wouldn't be airtight, so the pressure would equalize before any significant movement.
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