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## Homework Statement

"A ball with radius R and mass m turns around a horizontal axis through his center with an angular speed [tex]\omega_0[/tex]. In that condition the ball, without an initial velocity in the center, is put on top of a table. The friction coëfficient between the ball and the table is µ. Calculate the distance that the ball travels until it doesn't slip anymore.

([tex]I_{ball} = \frac{2}{5} m R^2[/tex])"

## Homework Equations

[tex]I_z \frac{\mathrm d \omega_z}{\mathrm d t} = \tau_z[/tex]

## The Attempt at a Solution

Hello,

I think I solved it ([tex]x = \frac{2R^2\omega_0^2}{49\mu g}[/tex]),

but there were no answers with the exercises, so I just want to check if I did the right conceptual steps:

So first of all, no matter how slow the ball was spinning, there will always be slipping at first, because v = 0 (when there's no subscript, I'm talking about the CM) and friction thus needs to deliver positive work (at the cost of kinetic energy, also producing heat, hence making an energy approach futile).

Now I thought, the necessary and sufficient condition for rolling without slipping is [tex]\omega R = v[/tex], so basically I just need to calculate how long it will take for v to RISE and omega to DROP until they reach this relation.

You can calculate v with the kinematic equation [tex]v^2 = 2ax[/tex], assuming [tex]F = \mu m g[/tex]. One gets an expression of v in function of the distance x.

We can also calculate the time derivative of omega (see "Relevant Equations") and then we use two other kinematic equations: one to find the time duration of slipping (in function of x) and then using that we use [tex]\omega = \omega_0 + \frac{\mathrm d \omega_z}{\mathrm d t} \Delta t[/tex], giving us omega in function of x.

We then plug both expressions into v = wR and solve for x.

One more concrete question:

This implies that the change in applied friction is abrupt, correct? Before the contact W = 0, then as soon as there is contact F = µmg, until v is reached at which point it goes abruptly from µmg to 0 again. There are no transient zones, correct?