Calculation of minimum angular velocity of a mass on a spinning plate

[gnits]

Homework Statement

How to calculate minumum angular velocity of a mass on a spinning plate

Relevant Equations

f=mrw^2

Problem Statement: How to calculate minumum angular velocity of a mass on a spinning plate
Relevant Equations: f=mrw^2

Hi, here's the question:

a) A rough horizontal plate rotates with a constant angular velocity of w about a fixed vertical axis. A particle of mass m lies on the plate at a distance of 5a/4 from the axis. If the coefficient of friction between the plate and the particle is 1/3 and the particle reamins at rest relative to the plate, show that w = sqrt(4g/15a).

b) The particle is now connected to the axis by a horizontal light elastic string, of natural length a and modulus 3mg. If the particle remains at rest relative to the plate and at a distance of 5a/4 from the axis, show that the greatest possible angular velocity of the plate is sqrt(13g/15a).

c) and find the least possible angular velocity.

I have done parts a) and b). It is part c) that I don't get.

I solved part b) by equating the frictional force + the tension in the elastic string = centripetal force (mrw^2)

Solving this I get the required answer of w = sqrt(13g/15a).

I understand that if w were greater than this then the particle would start to move further away from the axis.

But I don't see how w could be less than this and still have the particle 5a/4 from the axis and of course, I therefore don't see how I would calculate this.

Thanks for any help,
Mitch.

 

[TSny]

b) The particle is now connected to the axis by a horizontal light elastic string, of natural length a and modulus 3mg.

I'm not sure of the meaning of "modulus" for an elastic string. Is it the same as the elastic constant ("spring constant")? If so, shouldn't the value have dimensions of force per unit length? Did you mean to type 3mg/a instead of 3mg?

For part (c), it might help to consider what would happen if the plate is not rotating at all and you place the particle (with the elastic string) at a distance of 5a/4.

 

[gnits]

Hi TSny,

That helped a lot. Yes, I see now. When the plate is not rotating and the string is stretched then the frictional force is acting away from the axis. It all comes out easily then and I agree with the expected answer.

Thanks a lot,
Mitch.