A block placed in a horizontal hollow cylinder

[SciencyBoi]

Homework Statement

A block is placed inside a horizontal hollow cylinder. The cylinder is rotating with constant angular speed one revolution per second about its axis. The angular position of the block at which it begins to slide is 30° below the horizontal level passing through the center. Find the radius of the cylinder if the coefficient of friction is 0.6, What should be the minimum constant angular speed of the cylinder so that the block reach the highest point of the cylinder?

Homework Equations

Centrifugal force = mv² / r
Angular speed = Tangential speed/Radius
Friction = μN

The Attempt at a Solution

Picturing a cross section of the hollow cylinder rotating with an angular velocity of 2π radians/sec with forces marked;

Equating forces as the block can be assumed to be in rest (constant slipping);
(There would be no centripetal force because the block isn't moving with the cylinder, or isn't in a circular motion)
Mgcos(30) = μ[Mgsin(30)]
=> μ= √3
Which contradicts the statement made by the question that coefficient of friction is 0.6.
Also, this isn't giving any information about the radius as the radius doesn't even come in the equations.

However, according to the book, the solution is

The solution considers the ω of the block (equation1) to be the ω of the cylinder, when in actuality, the ω of the block is zero (it's at rest), and 5hus there shouldn't be any centrifugal forces acting on it.Please guide as to where I am wrong. I am sure 5to be missing some concept, because the question is correct. Thank you very much.

 

[TSny]

I think the problem wants you to assume that the block is placed at the bottom of the rotating cylinder such that the block is initially moving with the wall of the cylinder. So, initially the block is not slipping on the wall. The block moves with the wall without any slipping until it reaches the 30 degree position at which point slippage begins.

 

[haruspex]

in actuality, the ω of the block is zero (it's at rest),

The question says

at which it begins to slide

That is, it was rotating with the cylinder until the given angle was reached. The transition to sliding is determined by the balance of forces just before it started sliding.

 

[SciencyBoi]

The question says

That is, it was rotating with the cylinder until the given angle was reached. The transition to sliding is determined by the balance of forces just before it started sliding.

I have now understood it. It was a language problem apparently. Thank you very much.

 

[SciencyBoi]

So, initially the block is not slipping on the wall. The block moves with the wall without any slipping until it reaches the 30 degree position at which point slippage begins.

Thank you very much, I seem to have understood it now.