# Finding velocity given radius and coefficient of friction

## Homework Statement

A box with mass 2kg is on the edge of a circular platform of radius 6.0m. The coefficient of friction between the platform and the box is 0.3. The platform accelerates. Determine the speed when the box slips off the edge.

## Homework Equations

Fs = μsN
F⃗ net=ΣF⃗ =ma⃗
a(t) = d(vt)/dt
a(r) = v^2/r
a = [sqrt (ar^2) + (at^2)]

## The Attempt at a Solution

I know that this is non-uniform circular motion.
I don't think that what I'm doing is right.

ΣF(t) = -Ff = m(at)
at = - μs(g)
a(t) = - (0.3)(9.8)
a(t) = -2.94 m/s^2

ΣF(r) = m(ar)
ΣF(r) = (mv^2) / r

ΣF(net) = m * [sqrt (ar^2) + (at^2)]
Ff = m * [sqrt (ar^2) + (at^2)]
μs(g) = [sqrt (μs^2(g^2)) + (m^2*v^4) / r^2]
(μg)^2 = (μs^2(g^2)) + (m^2v^4) / r^2
((μg)^2 - μs^2(g^2)) /r^2 = (m^2*v^4)

So if I solve for v, I would take the fourth root of the left side after bringing m^2 over, and that doesn't seem right to me.

Delphi51
Homework Helper
I think you are working too hard. The acceleration is not given, suggesting you ignore F = ma. To my mind, it sounds like the condition for beginning to slip is simply when friction can no longer provide the necessary centripetal force to hold it on:
centripetal force = force of friction
which easily solves for v.

I think you are working too hard. The acceleration is not given, suggesting you ignore F = ma. To my mind, it sounds like the condition for beginning to slip is simply when friction can no longer provide the necessary centripetal force to hold it on:
centripetal force = force of friction
which easily solves for v.
How would that model non-uniform circular motion? Or does that not matter since the friction force is perpendicular?

Delphi51
Homework Helper
Strictly speaking, it does not apply to non-uniform circular motion where it could be that high tangential acceleration will cause the thing to slip into kinetic friction mode before centrifugal force overcomes the grip. If you have been doing problems like that in class, then do it with this one, too. Note that you will have to consider the combined radial and tangential forces and calculate when that exceeds the friction force.

But the wording suggests to me that the tangential acceleration is so small it can be ignored and slipping will occur in the radial direction.