# What is the moment of inertia for a water bottle?

• B
Hi, I'm setting up an equation for initial gravitational potential energy, and I'm just wondering what the moment of inertia, I, would be for a water bottle rolling down an inclined ramp?

Is it mr2 or ½mr2

Or would it be something else?

andrewkirk
Homework Helper
Gold Member
If the bottle is empty, it will be close to ##mr^2##. See the item here for 'thin cylindrical shell with open ends'.

But if the bottle contains water, whether completely or only partially full, it will be much more complicated. The water will mostly not rotate as the bottle rolls down. But there will be friction between the water and the bottle, which will cause some of the water to rotate, as well as generating heat and acting as a brake. I expect the rate of rolling would be somewhere between that for an object with MoI= ##mr^2## and one with MoI = ##\frac12 mr^2##.

phuntimes
Orodruin
Staff Emeritus
Homework Helper
Gold Member
Neither. A water bottle generally is not an idealised cylinder. It will depend a lot on the type of bottle and it is just impossible to give any sort of reasonable answer without more information.

sophiecentaur
Gold Member
2020 Award
Hi, I'm setting up an equation for initial gravitational potential energy, and I'm just wondering what the moment of inertia, I, would be for a water bottle rolling down an inclined ramp?

Is it mr2 or ½mr2

Or would it be something else?
Those two formulae refer to A hollow cylinder and A solid rod. Once you have realised what those two formulae actually refer to (and why), the whole thing should become clear. The answer will be somewhere in between for a real bottle. The geometry of the neck and the base will affect the value, too and so would the thickness of the envelope.
P.S. You can ADD moments of inertia.

jbriggs444
Homework Helper
Moment of inertia is a useful concept for bodies that rotate rigidly and accordingly have an unambiguous rotation rate. A bottle of water does not qualify.

Orodruin
Staff Emeritus