Cone rolling on a conical surface

[Saitama]

Homework Statement

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.

Homework Equations

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 $mg(\sin\alpha+(\omega \ell^2/g)\cos \alpha)$.

I don't have any idea about how would I start with b.

Any help is appreciated. Thanks!

 

[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 doesn't slip .

Hope this helps

 

[ehild]

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 $mg(\sin\alpha+(\omega \ell^2/g)\cos \alpha)$.

Is that square at the proper place ?

ehild

 

[Saitama]

Is that square at the proper place ?

ehild

No. Sorry about the typo. :)

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 doesn't slip .

Hope this helps

Thanks, I followed your suggestion and got the right answer.

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

 

[Tanya Sharma]

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

The cone rolls with slipping :)

 

[Saitama]

The cone rolls with slipping :)

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?

 

[haruspex]

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?

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.

 

[Tanya Sharma]

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?

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.

 

[Saitama]

Thank you haruspex and Tanya!