A perfectly stiff wheel cannot roll on a stiff floor?

[jmolmo]

A little "offtopic", but the question is "Will it roll or not?"

 

[DaveC426913]

Cool.

No, they will not roll.
The thread, the radii and the two touch-points of the wheels together form a trapezoid.
The first thing that rolling would theoretically result in is the lengthening of the longest side (the line between the two touch-points).
That trapezoid cannot be deformed and still maintain symmetry.
Not so sure now.

 

[haruspex]

Cool.

No, they will not roll.
The thread, the radii and the two touch-points of the wheels together form a trapezoid.
The first thing that rolling would theoretically result in is the lengthening of the longest side (the line between the two touch-points).
That trapezoid cannot be deformed and still maintain symmetry.

Well, it does depend on the relationship between the ratio of the radii and the angle of the slope. What if the slope increases to 60 degrees?

 

[kuruman]

Well, it does depend on the relationship between the ratio of the radii and the angle of the slope. What if the slope increases to 60 degrees?

To elaborate, static equilibrium is maintained when the torque due to gravity and the torque due to the tension in the string are in opposite directions. The former is non-zero as long as $\theta < 90^o$. The latter becomes zero when the radius of the inner wheel is large enough so that the position vector from the contact point to the point of application of the tension is horizontal. This happens when $r/R=\cos\theta$. For $r/R>\cos\theta$ the two torques are in the same direction and there can be no equilibrium. Note: Torques are calculated using the point of contact of the wheel with the incline as origin.

 

[Baluncore]

The distance between circle centres is the length of tight string.
It is not the change in height, Sin(30°), but the change in horizontal separation, Cos(30°) that is important.

They will not roll because R/2R = 0.5 string released, is less than Cos(30°) = 0.886 string required between centres.
Roll becomes possible when Cos(slope) = R/2R = 0.5 which will be when the slope becomes > 60°.

 

[A.T.]

Cool.

No, they will not roll.
The thread, the radii and the two touch-points of the wheels together form a trapezoid.
The first thing that rolling would theoretically result in is the lengthening of the longest side (the line between the two touch-points).
That trapezoid cannot be deformed and still maintain symmetry.
Not so sure now.

View attachment 223687

It boils down to whether to string goes above or below the contact points.

 

[A.T.]

Cool.

Can this spool roll down the incline without slipping, as the problem statement suggests?

Found here:
http://www.chegg.com/homework-help/questions-and-answers/spool-move-spool-rests-top-incline-made-two-uniform-disks-radius-r-mass-m-connected-horizo-q24458961

 

[Baluncore]

Can this spool roll down the incline without slipping, as the problem statement suggests?

No. But that is not a genuine solvable problem. It is an advertisement for “advanced physics tutors”. It is designed to disempower and confuse students to the point where they subscribe to a study program. Now being advertised on PF.

 

[jmolmo]

Can this spool roll down the incline without slipping, as the problem statement suggests?

View attachment 223703

Found here:
http://www.chegg.com/homework-help/questions-and-answers/spool-move-spool-rests-top-incline-made-two-uniform-disks-radius-r-mass-m-connected-horizo-q24458961

But, if I pull the string upwards...? Will it roll?

 

[jmolmo]

I promised some pictures earlier, and I've finally made them. Here are three pictures, showing a wheel rotating around a point of contact with the ground. The first is a perfectly circular wheel, the second is a wheel with the bottom flattened (eg a tyre compressing under the weight of the load) and the third is a polygon.

We see that, if the wheel rotates around the foremost point of contact in the direction of travel (marked) through a non-zero angle, it all works OK for the polygon and the flattened wheel, but not for the perfect circle, which has to go below the floor.

The answer arrived at in the thread above is that there is no rotation through any angle about that point. Rather, the statement that the wheel is rotating about that point is just a description of the instantaneous relative velocities of points on the wheel.
View attachment 215189 View attachment 215190 View attachment 215191

You're right ... In a rigid body, you can't consider 2 simultaneous "centers of rotation" at the same time (so, a point from which all the other body points motion can be described as circles around that "point")... such body would be no rigid.

The only way you can describe the movement of the points of a wheel if the geometrical lowest point of the wheel is moving along a line, is as the composition of one translation movement that follows that "line", and one rotational movement around the center of the wheel. So, or that "center of rotation" is constantly flipping between the center of the wheel and the geometrical contact point (but never both at the same time), or there's no point in saying that the contact point has been the center of rotation at any time, even if an infinitesimal time is considered.

 

[Baluncore]

But, if I pull the string upwards...? Will it roll?

Define upwards, and what you mean by pull. Consuming string at the fixed anchor point will move the spool up the ramp. Feeding string out from the fixed anchor point will allow the spool to roll. Raising the anchor point to change the angle between the string and the slope will make a big difference at some point.

 

[jmolmo]

Define upwards, and what you mean by pull. Consuming string at the fixed anchor point will move the spool up the ramp. Feeding string out from the fixed anchor point will allow the spool to roll. Raising the anchor point to change the angle between the string and the slope will make a big difference at some point.

If it's an string, I think it's clear what I mean by pull. If you mean how much pull, then image you do very, very little at first (0.01N) ... and then more and more.