# Cone rolling on a conical surface

1. Jun 10, 2013

### Saitama

1. The problem statement, all variables and given/known data
A round cone A of mass $m$ and half-angle $\alpha$ rolls uniformly and without slipping along a round conical surface B so that its apex O remains stationary. The centre of gravity of the cone A is at the same level as point O and at a distance $\ell$ from it. The cone's axis moves with angular velocity $\omega$. Find:
a)the static friction force acting on the cone A
b)at what values of $\omega$, the cone A will roll without slipping if coefficient of friction between the surfaces is equal to k.

2. Relevant equations

3. The attempt at a solution
Part a is easy to solve. I solved it by making an FBD of the cone A. Hence, force due to friction is $mg(\sin\alpha+(\omega \ell^2/g)\cos \alpha)$.

Any help is appreciated. Thanks!

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2. Jun 10, 2013

### Tanya Sharma

Hi Pranav

Since you have solved the first part, second one should be easy for you

The maximum static friction between the surfaces is kN.The force of friction you have calculated in part A should be less than kN .Find value of N from FBD ,put in the relation and you get values of ω for which the cone doesnt slip .

Hope this helps

3. Jun 11, 2013

### ehild

Is that square at the proper place ?

ehild

4. Jun 11, 2013

### Saitama

No. Sorry about the typo. :)

But what if $\omega$ becomes greater than required? How would it affect the motion of cone?

5. Jun 11, 2013

### Tanya Sharma

The cone rolls with slipping :)

6. Jun 11, 2013

### Saitama

I still don't get it. If $\omega$ becomes greater, the force of friction would be greater than kN. But kN is the maximum friction force, how is it possible?

7. Jun 11, 2013

### haruspex

If $\omega$ becomes greater, the force of friction needed to roll without slipping would be greater than kN. Therefore it will slip, so this greater value of ω is beyond the range.
Note that the question asks not for a single value of ω but for a range of values.

8. Jun 11, 2013

### Tanya Sharma

The maximum friction will be kN .For values of ω,which give value of friction greater than the limiting value ,the cone rolls with slipping .In other words ,values of friction beyond the limiting value is not feasible,hence irrelevant .It cannot increase beyond a certain limit .

The relation between friction and angular velocity which you have obtained is applicable only upto the limiting value i.e kN .

Its the same thing which we do in maths ,like x=3y ,y=N(i.e 1,2,3...),x≤18 .x will be multiple of 3 but the maximum value x can take is 18.

Last edited: Jun 11, 2013
9. Jun 11, 2013

### Saitama

Thank you haruspex and Tanya!