Maximum radius given rps, static and kinetic friction coefficients

[tehnexus]

Homework Statement

A computer scientist has dropped a tiny pizza crumb onto a magnetic disk. Later, when it's turned on, the disk is rotating at 88 revolutions per second about its axis, which is vertical. The coefficients of kinetic and static friction between pizza crumbs and the disk are 1.3 and 1.5 respectively. What is the maximum radius from the axis at which a pizza crumb will stay on the disk?

Homework Equations

F=ma, a=v^2/r, v=rω

The Attempt at a Solution

i really am not sure how to do this question, am i supposed to equate the centripetal force on the crumb with the frictional force? i have converted the rps to rad/s which is 552.93. please help me!

 

[RedRook]

There are a couple of things to consider here. The first thing is understanding the forces on the object.
$F_M = \mu_s * F_N$
$F_M =$ Maximum force of friction
$\mu_s =$ Static coefficient of friction
$F_N =$ Normal force

Static friction doesn't always produce a force. It only produces a force up to a maximum in opposition to motion on a plane. Now you already mentioned centripetal force. This is the force that is required to make something rotate. If you don't have enough centripetal force, then you don't have a solid orbit. Normal force is perpendicular to the plane of motion. Can you think of any forces that are perpendicular to the plane of motion? Basically what we need to find is what horizontal centripetal force exceeds the perpendicular normal force times the coefficient.

F = ma

We don't know the mass, but we know the forces in question are applied to the same mass, so what matters is the acceleration. If you can figure out an acceleration acting perpendicular to the disc, then you should be able to find the maximum horizontal acceleration toward the center that can occur with the static coefficient of friction.

 

[tehnexus]

So the kinetic friction coefficient does not affect the question? but i understand what you are saying thanks.

 

[RedRook]

Correct. Static friction is a ratio that represents how well two things are stuck together. It is all that applies until we have movement between the objects. Once we have movement you kind of get a skipping motion on the microscopic level. Think of it like two pieces of metal that have fitting triangular teeth on their surface. If you press against them and push, they will stay together. If you ever managed to push hard enough the teeth would start to skip against each other, and the amount of resistance in that skipping is less than when they fully fit together. That is why we need to numbers. We need one for when things are fully fit together and move as one, and one for when they start to slip and skip against each other.

 

[HalfLostAndSearching]

I would approach this problem by first understanding the physical principles involved. In this case, we have a rotating disk and a tiny pizza crumb that is subject to both static and kinetic friction.

To determine the maximum radius at which the crumb will stay on the disk, we can use the equation F=ma, where F is the net force acting on the crumb, m is the mass of the crumb, and a is the acceleration of the crumb. In this case, the net force on the crumb is the difference between the centripetal force (mv^2/r) and the frictional force (μmg), where μ is the friction coefficient, m is the mass of the crumb, g is the acceleration due to gravity, and r is the radius.

Setting the net force equal to zero, we can solve for the maximum radius:

mv^2/r - μmg = 0

r = mv^2/μmg

Now, we need to determine the value of the velocity (v) at the maximum radius. We can use the equation v=rω, where ω is the angular velocity of the disk. Substituting this into the equation above, we get:

r = (rω)^2/μmg

Solving for r, we get:

r = √(μmg/ω^2)

Now, we can substitute the given values into this equation to get the maximum radius at which the crumb will stay on the disk:

r = √(1.3*9.8/552.93^2) = 0.00016 meters

Therefore, the maximum radius at which the crumb will stay on the disk is approximately 0.00016 meters.