Friction coefficient on a rotating cylinder [mechanics]

[albatross84]

Homework Statement

Here is a picture of the problem I am trying to solve in a book. The problem I am having trouble with is #4 (the solution is provided in the brackets below the question -> 0.432)
http://img15.imageshack.us/img15/241/questiona.jpg

Homework Equations

F=ma
Ff=uma
etc...

The Attempt at a Solution

I have tried accounting for all the possible cases (including all free body diagrams I could think of) - Fg, and two Fn's and Ff's on both sides of contact (as well as individually). I have drawn all of my Fn's and Fg's through the center of gravity, which I am not entirely sure about. All of my attempts have so far given me a different number from the suggested solution.
In particular, I was hoping someone could help me out with the present forces and how they are drawn on the free body diagram (center of gravity?). Thanks.

 

[tiny-tim]

Welcome to PF!

Hi albatross84! Welcome to PF!

… I have drawn all of my Fn's and Fg's through the center of gravity, which I am not entirely sure about …

Yes, you're right not to be sure …

reaction forces always go through the point of contact (so in this case they'll also go though C) …

only the weight goes through the centre of mass

 

[nlightner]

I would approach this problem by first understanding the concept of friction and its coefficient. Friction is a force that opposes the motion of an object and is dependent on the surface materials and the normal force acting on the object. The friction coefficient is a dimensionless quantity that represents the ratio of the friction force to the normal force.

In this problem, a rotating cylinder is in contact with a horizontal surface, and we are asked to find the friction coefficient. To solve this problem, we need to consider the forces acting on the cylinder. The only force acting on the cylinder in the horizontal direction is the friction force, which is equal but opposite to the force applied by the surface. This means that the friction force is the only force responsible for the rotation of the cylinder.

To determine the friction coefficient, we need to use the equation Ff=uma, where Ff is the friction force, u is the friction coefficient, m is the mass of the cylinder, and a is the acceleration of the cylinder. In this case, the acceleration of the cylinder is related to its angular velocity, and we can use the equation a=rα, where r is the radius of the cylinder and α is the angular acceleration.

Now, we can write the equation for the friction force as Ff=umrα. Since we are given the angular velocity and the radius of the cylinder, we can calculate the angular acceleration using the equation α=ω^2/r, where ω is the angular velocity. Substituting this into the equation for the friction force, we get Ff=umrω^2.

Finally, we can use the equation F=ma to relate the friction force to the normal force acting on the cylinder. Since the cylinder is not moving in the vertical direction, the normal force must be equal to the weight of the cylinder, which is mg. Substituting this into the equation for the friction force, we get Ff=umrω^2=mg. Solving for the friction coefficient, we get u=g/(rω^2).

In conclusion, the friction coefficient on a rotating cylinder is dependent on the surface materials and the angular velocity of the cylinder. As a scientist, it is important to understand the concept of friction and how it affects the motion of objects. By applying the relevant equations and considering all the forces acting on the object, we can solve problems like this one and gain a better understanding of the physical world around us