Angular velocity using Newton's law

[Huumah]

Homework Statement

Small object of mass m is placed on the inner surface of the conical dish which can rotate with angular speed ω.

For μ=0.3 calculate the minumum and the maximum ω such that the object is remains at rest. Solve the problem using the indicated axes x-y.

Do the following steps:
(a) Show all forces acting on the object and and the acceleration
(b) Write Newton’s second law along both axes and solve.

Homework Equations

F=ma=m*ω²*r

The Attempt at a Solution

I have plugged in numbers and got answers for ω_max and ω_min but I don't get the right answer.
I have solved it right when I choose axis the ordinary X-Y axis but I need to know how to solve with using the other axis mentioned in the problem

The right answer is
ω_max=7.214 rad/s and ω_min =3.405 rad/s

 

[Saitama]

Your equation for both the X and Y direction is wrong.
You have taken the wrong component of mg and mω²r along the X direction. In the RHS, it won't be completely mω²r, it will be some component of this force. Make a free body diagram.

 

[Huumah]

In my free body diagram i have the fricton coefficient on both sides because it can go both upwards and downwards.

I don't really get what I should do on the right hand side for mω²*r

I have been trying for a few hours now and I am completely lost now.

 

[Saitama]

In my free body diagram i have the fricton coefficient on both sides because it can go both upwards and downwards.

Correct!

I don't really get what I should do on the right hand side for mω²*r

The force mω²r doesn't act in the X direction, so you will have to take its component along the X direction and solve.
See the image below, that should help you.

 

[Huumah]

Thanks very much.

I did not know in what direction the ω² force was heading. Tried to read it in my physics book but english is not my first language so i didn't understand it.

I feel like I'm getting closer but there is something wrong in my 2 equations because I can't solve them ( because there is something wrong with them)

Can you give my a hint, Sir?

 

[Doc Al]

I did not know in what direction the ω² force was heading. Tried to read it in my physics book but english is not my first language so i didn't understand it.

ω²r is the centripetal acceleration. Don't think of mω²r as a force, but as mass*acceleration. (View things from the usual inertial frame.)

I feel like I'm getting closer but there is something wrong in my 2 equations because I can't solve them ( because there is something wrong with them)

Why can't you solve them? Eliminate N. (I think you have one of your angles wrong.)

 

[Huumah]

I always got an error when i tried to solve it using wolfram alpha.

I just probably had a silly typo because it worked when I typed it in again.

I got it right when i changed the angle to 30° in the formula for y axis.

But I still don't get why it is not 60° in the y-axis formula.

Anyway thank you both so much for the help. Such a lovely community.

 

[Doc Al]

I got it right when i changed the angle to 30° in the formula for y axis.

But I still don't get why it is not 60° in the y-axis formula.

The angle between mg and your y-axis is 30 degrees; the angle between mω²r and your y-axis is 60 degrees. And sin(30)=cos(60) & sin(60)=cos(30).