Undergrad Mechanics of Rolling and Striking Cones

[EngineeringFuture]

TL;DR

Mechanics of rolling cones

Does an object have to be either spherical or cylindrical to be rolled in a straight line. Could an approximately conical object be rolled in a straight line or struck to roll in a straight line? A cone can be rolled in a straight line if different forces are applied at different spots, but once the force is no longer applied, can a cone keep rolling by its inertia alone?

I believe the answer is no, not if the apex of the cone remains on the surface, but I think there are unusual surfaces where a cone of non-uniform density where a cone would slightly deform so the apex would lift into the air and the cone would keep rolling.

Also, my terminology is terrible, so please excuse my terminology. There is one concept that I'm thinking about but can't express.

 

[Baluncore]

If the circumference of the roller varies, like a cone or frustum, and remains in frictional contact with the flat surface, then the friction will prevent free rolling motion in a straight line, on a flat surface.

 

[EngineeringFuture]

If the circumference of the roller varies, like a cone or frustum, and remains in frictional contact with the flat surface, then the friction will prevent free rolling motion in a straight line, on a flat surface.

Do you have any links I could see with the diagrams and math worked out?

 

[Baluncore]

Do you have any links I could see with the diagrams and math worked out?

This is more fundamental than mathematics.

Make a conical roller from a sheet of paper and some sticky tape, like a witches hat. Roll it in a straight line and watch how it slips.

If wheels of different circumference, rotate at the same rate, they will travel different distances along arcs of different radii.

When you drive a car around a corner, the wheels on either side rotate at different rates. That is why there is a differential gear.

Conical rollers are used in tapered roller bearings, where the path lengths are in proportion to the roller circumference.

 

[A.T.]

Could an approximately conical object be rolled in a straight line...

Sure, even uphill.

 

[sophiecentaur]

TL;DR Summary: Mechanics of rolling cones

A cone can be rolled in a straight line if different forces are applied at different spots,

If the cone is a solid material and the contact with the floor doesn't slip then that is not true.

Why? If the cone rotates once then the distance travelled (on a plane surface) by the part with least radius (r1) will be 2πr1 (the circumference) and the distance travelled by a part near the maximum radius (r2) will travel 2πr2. That implies the fat end travels further than the thin end. That isn't in a straight line. To force a straight line you have to let every part slip.

 

[EngineeringFuture]

If the cone is a solid material and the contact with the floor doesn't slip then that is not true.

Why? If the cone rotates once then the distance travelled (on a plane surface) by the part with least radius (r1) will be 2πr1 (the circumference) and the distance travelled by a part near the maximum radius (r2) will travel 2πr2. That implies the fat end travels further than the thin end. That isn't in a straight line. To force a straight line you have to let every part slip.

You're absolutely correct. I was an idiot above.