Can a cone roll in a straight line without slipping?

[hackhard]

<<Mentor note: Thread split from https://www.physicsforums.com/threads/torque-opposite-in-direction-to-change-in-angular-momentum.866882/>>

it can be proved that in this case pure rolling without slipping is impossible
ive assumed the cone to be right circular
proof by contradiction

https://drive.google.com/open?id=0B5m-h82V0Ej2b1g5YmFhSUk5UEE
https://drive.google.com/open?id=0B5m-h82V0Ej2aFRxR2RzRlFmaVk
https://drive.google.com/open?id=0B5m-h82V0Ej2aGxLRENyYmdXZVU

 

[hackhard]

And yet we observe cones, in Nature, rolling without slipping... perhaps this rolling is not "pure"?

we do observe but rolling without slipping about axis in post #1(z-axis) is impossible
here is 1 eg of cone rolling without slipping (without ext force)

the long line shows path of com
https://drive.google.com/open?id=0B5m-h82V0Ej2YlVSNlB2UVl2SE0

 

[Simon Bridge]

... please be clearer: you are not really saying that rolling a cone without slipping is impossible and yet it happens?
Note: diagram meaningless without labels. Shouldn't the cone com move in a circle? That is what happens with every cone I've rolled.

What about:
https://www.physicsforums.com/threads/lagrangian-mechanics-cone-rotating-over-a-plane.670424/
... in this case, though, the weight-friction couple is not enough to lift the cone off the surface.

How about:
https://www.physicsforums.com/threa...-its-side-without-slipping-on-a-plane.750264/

 

[hackhard]

Shouldn't the cone com move in a circle? That is what happens with every cone I've rolled

Lets assume all cones roll without slipping about z-axis thru apex
Lets now repeat this with cones of decreasing apex angle till it is close to 0
Object suddenly stops rotating about z-axis and rolls in straight path
this is a discontinuity in observation

 

[Simon Bridge]

No it isn't... we observe cones rolling in circles and we observe cylinders rolling in straight lines.
Are you, therefore, saying that a cone cannot roll in a circle with it's apex as the center?

 

[hackhard]

Are you, therefore, saying that a cone cannot roll in a circle with it's apex as the center?

no it cannot except if com is directly above apex (while rolling) (ie cone with a certain obtuse apex angle)
with no ext force object can rotate only about line thru com
cone can rotate (with rolling ) but only about line thru com

 

[hackhard]

the long line shows path of com
https://drive.google.com/open?id=0B5m-h82V0Ej2YlVSNlB2UVl2SE0

Shouldn't the cone com move in a circle?

it is moving in circle in the com inertial frame
com is moving with const velo

 

[Simon Bridge]

OK... be that as it may, how does that address the question?... allow a cone executing a circle by rolling without slipping so that the z axis is at the center of the circle. Its just not on the apex. The angular momentum of the cone about its symmetry axis is still changing direction... does this not imply an unbalanced torque?

Note. In the com frame, the com is stationary: that is what "com frame" means.

 

[hackhard]

at last i figured it out
my previous posts may be bit incorrect
rolling without slipping is always impossiblefor right circular cone
i had proved it for axis through apex in my previous post

it can be proved that in this case pure rolling without slipping is impossible
ive assumed the cone to be right circular
proof by contradiction

https://drive.google.com/open?id=0B5m-h82V0Ej2b1g5YmFhSUk5UEE
https://drive.google.com/open?id=0B5m-h82V0Ej2aFRxR2RzRlFmaVk
https://drive.google.com/open?id=0B5m-h82V0Ej2aGxLRENyYmdXZVU

a simpler proof--
for rolling without slipping
velocity of foot of altitude on base (P) must be = velocity of instantaneous point of contact (Q) furthest from apex
for any axis normal to ground surface , normal dist to P is not equals to normal dist to Q
so any curved motion is impossible (due to same velocity of 2 points at different distances from same axis)
translation is also impossible since the instantaneous point of contact just below P will have diff forward velo (due to equal angular speed and diff cross-section radius) (cross-section is normal to symmetric axis not to ground surface)

what we observe right circular cones rolling is actually cones slipping for small instants
when right circular cones is given push
near the apex friction acts backwards (ie cones slips forward near apex)
near the base friction acts forwards (ie cones slips backward near base)
this makes cones roll and rotate

however rolling without slipping is possible for oblique cone whose base is normal to ground surface
this can visualized by slicing the oblique cone normal to ground surface into very thin rings
consider only the slice at apex and that at the base
this can be related with a differential drive system
slice at apex remains at rest and oblique cones rotates about z-axis through apex
on static friction is required at apex

addressing initial ques angular p about z-axis is const . torque about z-axis is 0 , so it is balanced
angular p about other axis changes. torque about other axis= friction near apex * r , so it is balanced

 

[Orodruin]

we do observe but rolling without slipping about axis in post #1(z-axis) is impossible
here is 1 eg of cone rolling without slipping (without ext force)

the long line shows path of com
https://drive.google.com/open?id=0B5m-h82V0Ej2YlVSNlB2UVl2SE0

Please stop linking to google drive documents. Apart from forcing people to leave the site, there is no guarantee that external documents will be maintained.

Furthermore, your assertion that a cone rolling without slipping is impossible is directly false. The cone is a flat surface and can roll without slipping on a flat surface. This can be easily verified by looking at the area the cone would sweep on the surface, which is going to be diffeomorphic to the cone itself.

 

[hackhard]

Furthermore, your assertion that a cone rolling without slipping is impossible is directly false

please disprove this

a simpler proof--
for rolling without slipping
velocity of foot of altitude on base (P) must be = velocity of instantaneous point of contact (Q) furthest from apex
for any axis normal to ground surface , normal dist to P is not equals to normal dist to Q
so any curved motion is impossible (due to same velocity of 2 points at different distances from same axis)
translation is also impossible since the instantaneous point of contact just below P will have diff forward velo (due to equal angular speed and diff cross-section radius) (cross-section is normal to symmetric axis not to ground surface)

 

[Orodruin]

please disprove this

I just did, if you read my reply properly.

 

[hackhard]

what is wrong with my proof

a simpler proof--
for rolling without slipping
velocity of foot of altitude on base (P) must be = velocity of instantaneous point of contact (Q) furthest from apex
for any axis normal to ground surface , normal dist to P is not equals to normal dist to Q
so any curved motion is impossible (due to same velocity of 2 points at different distances from same axis)
translation is also impossible since the instantaneous point of contact just below P will have diff forward velo (due to equal angular speed and diff cross-section radius) (cross-section is normal to symmetric axis not to ground surface)

 

[Simon Bridge]

@hackhard
Perhaps see:
Jackman H. (2008) Rolling Constraints
http://www.ingvet.kau.se/juerfuch/kurs/amek/prst/07_roll.pdf
... page 8.

Jackman starts out much as you do but arrives at a different conclusion.
Working out the constraints for rolling without slipping of a right circular cone on a plane surface is a standard exercise for classical mechanics and engineering students.

 

[andrewkirk]

my previous posts may be bit incorrect
rolling without slipping is always impossiblefor right circular cone

It's easy to see why a cone of any angle can roll without slipping.

Let the angle between the cone wall and its axis be $a$. For a point $P$ on the cone that is distance $r$ from the apex, the cone circumference at that point is $2\pi r\sin a$. Say the cone is rolling at a constant angular velocity of $\omega$ radians per sec relative to its (precessing) axis. Then as point $P$ touches the ground it has linear speed $\omega r\sin a$ relative to the cone's COM. Say the cone is rotating around the origin at rate $\beta$. THen the speed of that part of the cone relative to the ground is $\beta r$. So to have no slipping we require the two to be equal, which requires $\beta r=\omega r\sin a$, that is $\beta=\omega\sin a$. Since $r$ is not part of this equation, it is possible for it to hold at all radii, so the cone can roll without slipping.

 

[Simon Bridge]

It's quite tricky to figure out what you are saying since you are unclear in your definitions.
But I'll give it a go since the other disproofs provided you do not show you where your own reasoning has failed you.
I cannot be sure - but I can try to point at places that you can check for yourself.

a simpler proof--
for rolling without slipping
velocity of foot of altitude on base (P) must be = velocity of instantaneous point of contact (Q) furthest from apex

... I think more detail is needed here: it is unclear if you are mixing up reference frames or just assuming the end result. But I suspect this is primarily where you come unglued. In the x-y-z frame in your diagrams, the velocity of Q is zero (points of contact must have zero relative velocity to avoid slipping - hence "assuming the result" when you ruled this out.) Your own diagram shows P and Q having different velocities.

for anyaxis normal to ground surface , normal dist to P is not equals to normal dist to Q
so any curved motion is impossible (due to same velocity of 2 points at different distances from same axis)

All the points are also accelerating... I think you a imagining that the cone rolls like a row of vertical disks ... which is not the case.
A point on the surface of the cone traces out a tilted circle.

translation is also impossible since the instantaneous point of contact just below P will have diff forward velo (due to equal angular speed and diff cross-section radius) (cross-section is normal to symmetric axis not to ground surface)

Does this mean the cone cannot roll, without slipping, in a straight line?