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

In summary, the problem involves calculating the minimum and maximum angular velocity of a mass on a spinning plate at a distance of 5a/4 from the axis, with a coefficient of friction of 1/3 and a horizontal light elastic string. Part (a) is solved by equating the frictional force and tension in the string to the centripetal force. Part (b) is solved by finding the greatest possible angular velocity using the same equation. For part (c), the plate is not rotating and the string is stretched, resulting in a frictional force away from the axis. This leads to the minimum possible angular velocity being equal to sqrt(13g/15a).
  • #1
gnits
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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.
 
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  • #2
gnits said:
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.
 
  • #3
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.
 
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Related to Calculation of minimum angular velocity of a mass on a spinning plate

What is the "Calculation of minimum angular velocity of a mass on a spinning plate"?

The calculation of minimum angular velocity of a mass on a spinning plate is a physics problem that involves determining the minimum rotational speed at which a mass can be placed on a spinning plate without falling off.

What are the variables involved in this calculation?

The variables involved in this calculation include the mass of the object, the radius of the spinning plate, the coefficient of friction between the object and the plate, and the acceleration due to gravity.

How is the minimum angular velocity calculated?

The minimum angular velocity is calculated using the formula: ωmin = √(g * μ * r), where ωmin is the minimum angular velocity, g is the acceleration due to gravity, μ is the coefficient of friction, and r is the radius of the spinning plate.

What is the significance of calculating the minimum angular velocity?

Calculating the minimum angular velocity is important in understanding the relationship between rotation, friction, and gravity. It also helps to determine the stability of objects placed on a spinning surface.

How does the minimum angular velocity change with different variables?

The minimum angular velocity is directly proportional to the radius of the spinning plate and the acceleration due to gravity. It is also affected by the coefficient of friction, with a higher coefficient resulting in a lower minimum angular velocity.

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