Rolling without slipping and coefficient of friction

[henry3369]

Homework Statement

A hollow spherical shell with mass 2.50kg rolls without slipping down a slope that makes an angle of 33.0 with the horizontal.
Find the minimum coefficient of friction μ needed to prevent the spherical shell from slipping as it rolls down the slope.

Homework Equations

The Attempt at a Solution

I already found force of friction = 5.34N.

It seems that I can get the minimum by setting force of friction = μN
So:
5.34 = μmgcos(Θ)

What I don't understand is why this is the minimum of coefficient of friction. How can one assume that this value is the minimum coefficient of friction to prevent slipping?

 

[ehild]

Homework Statement

A hollow spherical shell with mass 2.50kg rolls without slipping down a slope that makes an angle of 33.0 with the horizontal.
Find the minimum coefficient of friction μ needed to prevent the spherical shell from slipping as it rolls down the slope.

Homework Equations

The Attempt at a Solution

I already found force of friction = 5.34N.

How did you get that value?

It seems that I can get the minimum by setting force of friction = μN
So:
5.34 = μmgcos(Θ)

What I don't understand is why this is the minimum of coefficient of friction. How can one assume that this value is the minimum coefficient of friction to prevent slipping?

During rolling, it is the force of static friction that prevents slipping, (relative motion of the surfaces in contact). The static friction is not a defined value. It has an upper limit μN: F(static) ≤ μN.

 

[Pierce610]

You get the minimun coefficient at the equilibrium of two forces: one which accelerates the sphere down the plane with a reduct acceleration, the other which exists only as rection to the first, that is the friction force.
So first calculate the force with "reducted" g and then let it equal to the friction force.
I can anticipate you that when you'll simplified the two member of the equation to obtain the result, you will discovered that the coefficient is function only of the angle and not of the mass or the forces.
The minimun means only that it is enough to keep the mass motionless, and then the force that accelerate it should be greater of the friction force.
Verify the calculate: I obtain for the force a different value...