How far can the ant walk before slipping off the record?

[cdornz]

Homework Statement

A 1-mg ant is located near the center of a horizontal record (radius = 7 inches) which is rotating at 78rpm. The coefficient of friction between the ant and the record is 0.7. How far out towards the edge of the record can the ant walk before it will slip?

Homework Equations

circumference = d$\Pi$
ΔV = circumference * rpm
F_centripetal = m(v²/r)
$\Sigma$F=ma

The Attempt at a Solution

I got all of my numbers into lowest form/metric units.
So the radius is 0.1778 meters; rpm is now 78/60 or 1.3 rev/sec
Circumference = 0.3556$\Pi$; circumference = 1.11 meters
Velocity = 1.11/1.3; velocity = 1.45 m/s

I then listed the forces involved: F_centripetal; F_n; F_gravity; F_friction

I setup my free body diagram like this picture attached

I can then do the forces in the x direction (cosine) and the forces in the y direction (sin).
My question for now, is if I get a number for x and y, how will that tell me how long until the ant slips? We've never covered a question like this in my physics class before.

 

[voko]

Sorry, I can't decipher your diagram.

What is the condition that the ant NOT slip, i.e., keep going in circles?

 

[cdornz]

Wouldn't the centripetal force be keeping the ant going in circles? Since that force would be radially inward?

 

[cdornz]

sorry it keeps rotating.

 

[voko]

Yes, but WHAT creates the inward force keeping the ant in circular motion?

 

[cdornz]

Would it be that the ant is constantly being accelerated toward the center of the record?

 

[voko]

The ant is being accelerated BY the inward force. But HOW does the force originate?

Imagine the ant is not on a record, but is tied to an end of a horizontal rod rotating about the other end. WHAT keeps the ant rotating with the rod?

 

[cdornz]

I would think an unbalanced force consisting of it's weight, normal force, and friction. I went back to look at my notes on this section and all I have states an equation:

M*(N2∏)²)R/T²

 

[voko]

For an ant tied to a rotating rod, the source of the inward pull is the tension in rod/rope. But for an ant on a rotating surface with friction, what would that be?