# Which way will the conical object move?

1. Apr 3, 2016

### Vibhor

1. The problem statement, all variables and given/known data

2. Relevant equations

v = ωr

3. The attempt at a solution

Honestly speaking I have very little idea about the problem . I am not understanding the setup clearly . What role does the rails play while the cones move on them .Are the cones fixed to the rails ? Does the tuning of the object -right or left , means rotation of the object clockwise –anticlockwise or it means shifting of the object as a whole towards left or right as it moves forward ?

I would really appreciate if somebody could help me with the problem .

Many Thanks

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Last edited: Apr 3, 2016
2. Apr 4, 2016

### haruspex

I agree the question is far from clear. I shall assume the rails are horizontal, that the picture is the plan view ( looking down from above), that the rails are not parallel, that the dashed line is perpendicular to CD, and that the roller rolls without slipping.
Suppose the roller goes straight. After rotating through some angle, consider how far its points of contact have moved.

3. Apr 4, 2016

### Vibhor

Hello ,

I agree .

If 'r' is the radius of the cones and angle turned is θ , then the points of contacts have moved by $rθ$ .

4. Apr 4, 2016

### haruspex

But the radius where the cone touches rail AB is reducing, no?

5. Apr 4, 2016

### Vibhor

This is exactly where I am having trouble understanding ( the motion of the cone ) . So you are suggesting that the cones do not touch the floor ,instead they rest on the rails such that the the distance of the point , where the left cone touches the rail AB from mid point O , decreases as the object moves forward ??

So , now how should I use this fact to proceed further ?

6. Apr 4, 2016

### haruspex

How does that affect the forward movement of that cone compared with the other (assuming both cones turn through the same angle, without slipping)?

7. Apr 4, 2016

### Vibhor

The right cone moves faster as compared to left ??

8. Apr 4, 2016

Yes.

9. Apr 4, 2016

### Vibhor

Is my understanding in first part of post#5 correct ?

10. Apr 4, 2016

### haruspex

Yes.

11. Apr 4, 2016

### Vibhor

Does that mean the object as a whole translates forward and also rotates anticlockwise (as seen from the top) ??

12. Apr 4, 2016

### haruspex

As viewed from above, yes.

13. Apr 4, 2016

### Vibhor

and this is equivalent to object turning left i.e option (4) is correct ??

Last edited: Apr 4, 2016
14. Apr 4, 2016

### haruspex

Yes.

15. Apr 4, 2016

### Vibhor

Fantastic analysis of the problem .

Thanks a lot haruspex .

16. Apr 12, 2016

### Vibhor

Hi ,

I still have some niggling doubts .

What role does friction play in this problem ?

17. Apr 12, 2016

### haruspex

You have to assume some friction or the system would be unstable right from the start. Do you see why? Draw a diagram in the vertical plane. If the double cone were to slide slightly to one side, what would happen to its mass centre... up or down or the same?

18. Apr 12, 2016

### Vibhor

As the left cone moves forward , the radius decreases . As a result vL(speed of CM of left cone) > ωr . There is a tendency to slip forward and static friction acts backward . This increases ω .

Now , in case of right cone vR(speed of CM of left cone) < ωr . There is a tendency to slip backward and static friction acts forward.

The result of the two friction forces is to rotate the system anti - clockwise .

Does it make sense ?

19. Apr 12, 2016

### haruspex

Yes, I believe that works. An equivalent model is two wheels of different sizes fixed to the same shaft (and locked to it) on a frictional floor.

20. Apr 12, 2016

### Vibhor

Ok .

But would you agree that this is essentially the same reasoning we had discussed earlier in the thread ( your proposed way of looking at the problem ) . I mean in the earlier reasoning also , static friction was also involved although we didn't mention it ??